Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1607.05837

Dispelling Rayleigh's Curse

Jaroslav Řeháček, Martin Paùr, B. Stoklasa, L. Motka, Z. Hradil, Luis Lorenzo Sánchez-Soto

Published Jul 20, 2016• 3 citations
arXivHTMLPDF

Abstract

We devise a systematic method to determine the Fisher information required for resolving two incoherent point sources with a diffraction-limited linear imaging device. The resulting Cramér-Rao bound gives the lowest variance achievable for an unbiased estimator. When only intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect which has been dubbed as Rayleigh's curse. However, this curse can be lifted using suitable coherent measurements. In particular, we determine a class of measurements that can be easily implemented as projections on the orthogonalised derivatives of the point spread function of the system.

DOI
10.48550/arxiv.1607.05837

Related papers

Quantum-Limited Time-Frequency Multiparameter Estimation through Mode-Selective Photon Measurement

Vahid Ansari, John M. Donohue, Jano Gil-López +7 more

In optics, when attempting to find the distance separations s between two incoherent emitters, it is standard practice to take a full image and extract the single parameter s from it. In such scenarios, the smallest precisely measurable separation is limited by the point-spread function on the image plane, with a vanishingly small amount of information available for smaller and smaller separations especially when photon shot noise is the dominant noise source (as in, for example, astronomical observations). This limit can be formalised by the Fisher information, which quantifies the amount of information accessible per photon detection and is directly associated with Cramér-Rao lower bound. For direct intensity imaging, the Fisher information drops to zero for object separations smaller than the spread of the optical field, sometimes referred to as the Rayleigh's curse, which limits the usefulness of photon counting for metrology.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 1, 2019•0 citations

Resurgence of Rayleigh’s curse in the presence of partial coherence

Walker D. Larson, Bahaa E. A. Saleh

The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.

Optica•Oct 24, 2018•73 citations

Achieving quantum-limited optical resolution

Martin Paùr, B. Stoklasa, Z. Hradil +2 more

We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh precision.

arXiv:1606.08332•Jun 27, 2016•5 citations

Beating Rayleigh’s Curse by Imaging Using Phase Information

Weng-Kian Tham, Hugo Ferretti, Aephraim M. Steinberg

Every imaging system has a resolution limit, typically defined by Rayleigh's criterion. Given a fixed number of photons, the amount of information one can gain from an image about the separation between two sources falls to zero as the separation drops below this limit, an effect dubbed "Rayleigh's curse." Recently, in a quantum-information-inspired proposal, Tsang and co-workers found that there is, in principle, infinitely more information present in the full electromagnetic field in the image plane than in the intensity alone, and suggested methods for extracting this information and beating the Rayleigh limit. In this Letter, we experimentally demonstrate a simple scheme that captures most of this information, and show that it has a greatly improved ability to estimate the distance between a pair of closely separated sources, achieving near-quantum-limited performance and immunity to Rayleigh's curse.

arXiv:1606.02666•Physical Review Letters•Feb 15, 2017•175 citations

Generalization of Rayleigh's criterion on parameter estimation with incoherent sources

Lijun Peng, Xiao-Ming Lu

The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.

arXiv:2011.07897•Physical Review A•Apr 1, 2021•9 citations

Temporal Resolution of Partially Coherent Sources

Syamsundar De, Jano Gil-López, Benjamin Brecht +4 more

In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 21, 2021•0 citations
We devise a systematic method to determine the Fisher information required for resolving two incoherent point sources with a diffraction-limited linear imaging device. The resulting Cramér-Rao bound gives the lowest variance achievable for an unbiased estimator. When only intensity in the image plane is recorded, this bound diverges as the separation between the sources tends to zero, an effect which has been dubbed as Rayleigh's curse. However, this curse can be lifted using suitable coherent measurements. In particular, we determine a class of measurements that can be easily implemented as projections on the orthogonalised derivatives of the point spread function of the system.
In optics, when attempting to find the distance separations s between two incoherent emitters, it is standard practice to take a full image and extract the single parameter s from it. In such scenarios, the smallest precisely measurable separation is limited by the point-spread function on the image plane, with a vanishingly small amount of information available for smaller and smaller separations especially when photon shot noise is the dominant noise source (as in, for example, astronomical observations). This limit can be formalised by the Fisher information, which quantifies the amount of information accessible per photon detection and is directly associated with Cramér-Rao lower bound. For direct intensity imaging, the Fisher information drops to zero for object separations smaller than the spread of the optical field, sometimes referred to as the Rayleigh's curse, which limits the usefulness of photon counting for metrology.
The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.
We establish a simple method to assess the quantum Fisher information required for resolving two incoherent point sources with an imaging system. The resulting Cramér-Rao bound shows that the standard Rayleigh limit can be surpassed by suitable coherent measurements. We explicitly find these optimal strategies and present a realization for Gaussian and slit apertures. This involves a projection onto the optimal bases that is accomplished with digital holographic techniques and is compared with a CCD position measurement. Our experimental results unequivocally confirm unprecedented sub-Rayleigh precision.
Every imaging system has a resolution limit, typically defined by Rayleigh's criterion. Given a fixed number of photons, the amount of information one can gain from an image about the separation between two sources falls to zero as the separation drops below this limit, an effect dubbed "Rayleigh's curse." Recently, in a quantum-information-inspired proposal, Tsang and co-workers found that there is, in principle, infinitely more information present in the full electromagnetic field in the image plane than in the intensity alone, and suggested methods for extracting this information and beating the Rayleigh limit. In this Letter, we experimentally demonstrate a simple scheme that captures most of this information, and show that it has a greatly improved ability to estimate the distance between a pair of closely separated sources, achieving near-quantum-limited performance and immunity to Rayleigh's curse.
The basic idea behind Rayleigh's criterion on resolving two incoherent optical point sources is that the overlap between the spatial modes from different sources would reduce the estimation precision for the locations of the sources, dubbed Rayleigh's curse. We generalize the concept of Rayleigh's curse to the abstract problems of quantum parameter estimation with incoherent sources. To manifest the effect of Rayleigh's curse on quantum parameter estimation, we define the curse matrix in terms of quantum Fisher information and introduce the global and local immunity to the curse accordingly. We further derive the expression for the curse matrix and give the necessary and sufficient condition on the immunity to Rayleigh's curse. For estimating the one-dimensional location parameters with a common initial state, we demonstrate that the global immunity to the curse on quantum Fisher information is impossible for more than two sources.
In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.