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

Physical Review A

Quantum-statistical analysis of superresolution for optical systems with circular symmetry

Vladislav N. Beskrovnyy, Mikhail I. Kolobov

Published Oct 28, 2008• 17 citations
Publisher

Abstract

We present a detailed quantum-statistical analysis of superresolution for optical systems with circular symmetry based on our previous paper [Opt. Commun. 264, 9 (2006)].We consider as example an imaging system with two Fourier lenses; however, our approach can be easily generalized for more complex paraxial systems with circular symmetry. We present an original and very effective algorithm for numerical evaluation of the circular prolate spheroidal functions, which are the eigenmodes of our system. We investigate in detail the two-dimensional reconstruction point-spread function and provide examples of superresolution for rotationally symmetric and angularly dependent objects. Finally, we investigate the role of the local quantum fluctuations in the superresolution process and evaluate the local noise figure for the reconstruction process.

DOI
10.1103/physreva.78.043824

Related papers

Quantum fluctuations in superresolving Fourier-microscopy

Pierre Scotto, Pere Colet, Mikhail I. Kolobov +1 more

A new scheme, superresolving Fourier-microscopy is proposed, which detects the spatial Fourier spectrum of the image created by an additional Fourier lens. This procedure amounts in directly imaging on the observation plane the field coming through the pupil, and all the field transmitted by the pupil is again concentrated on a finite region of space. Using decomposition on the basis of the prolate spheroidal functions, object reconstruction is shown to be in principle possible. The gain in superresolution achieved in this scheme over the classical Rayleigh limit is investigated numerically.

European Quantum Electronics Conference (EQEC)•Jan 1, 2003•0 citations

Quantum limits of superresolution for imaging discrete subwavelength structures

Mikhail I. Kolobov

We present the quantum theory of superresolution for discrete subwavelength structures. It allows to formulate, in particular, the standard quantum limit of superresolution achieved for illumination of the structure by light in coherent state. Our theory is based on discrete prolate spheroidal sequences and functions which are the proper basis set of the problem. We demonstrate that the superresolution factor is much higher for discrete structures than for continuous objects for the same signal-to-noise ratio. This result is a clear illustration of the crucial role of a priori information in superresolution problems.

Optics Express•Jan 1, 2008•7 citations

Optimal quantum superresolution for full distance between incoherent optical sources in two dimensions

Junyan Li, Shengshi Pang

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.

arXiv:2508.08049•Physical Review A•Feb 12, 2026•0 citations

Quantum limits of super-resolution in reconstruction of optical objects

Vladislav N. Beskrovnyy, Mikhail I. Kolobov

We investigate analytically and numerically the role of quantum fluctuations in reconstruction of optical objects from diffraction-limited images. Taking as example of an input object two closely spaced Gaussian peaks, we demonstrate that one can improve the resolution in the reconstructed object over the classical Rayleigh limit. We show that the ultimate quantum limit of resolution in such a reconstruction procedure is determined not by diffraction but by the signal-to-noise ratio in the input object. We formulate a quantitative measure of super-resolution in terms of the optical point-spread function of the system.

arXiv:quant-ph/0412120•Physical Review A•Apr 4, 2005•40 citations

Superresolution at the quantum limit beyond two point sources

Hari Krovi

Superresolution refers to the estimation of parameters of an image with an accuracy beyond standard classical techniques such as direct detection. In seminal work by Lu et al., a measurement to estimate the separation distance of two point sources (with a known centroid) was shown to achieve the quantum Cramer-Rao bound. This work made implicit use of reflection symmetry of the sources. Here we present a framework that uses more general symmetry in a constellation to construct a quantum measurement that achieves the quantum Cramer-Rao bound in estimation of parameters. We show how this technique can be used to estimate parameters simultaneously in symmetric point-source constellations with more than two point sources. In order to use symmetry explicitly, we make use discrete point spread functions in momentum space that maintain this symmetry. This framework allows us to use techniques from quantum computing such as Fourier transforms and linear optical circuits to implement the optimal measurement. To our knowledge, this is first work that shows for more than two point sources achievable quantum limits of estimation and modal transformations.

arXiv:2206.14788•Jun 29, 2022•1 citation

Quantum Limits on Optical Resolution

Mikhail I. Kolobov, Claude Fabre

We discuss the ultimate limit imposed by quantum fluctuations of light for resolution of fine details in optical images. For this purpose, we extend in the quantum domain the classical analysis of the object reconstruction, or superresolution, in terms of prolate spheroidal function basis. We derive the expression for ultimate resolution limit in the reconstructed object using an illumination of the full object plane by a multimode squeezed vacuum. We show that the gain in resolution using multimode squeezed light is maximum when the Shannon number of the imaging system is close to unity.

Physical Review Letters•Oct 30, 2000•178 citations
We present a detailed quantum-statistical analysis of superresolution for optical systems with circular symmetry based on our previous paper [Opt. Commun. 264, 9 (2006)].We consider as example an imaging system with two Fourier lenses; however, our approach can be easily generalized for more complex paraxial systems with circular symmetry. We present an original and very effective algorithm for numerical evaluation of the circular prolate spheroidal functions, which are the eigenmodes of our system. We investigate in detail the two-dimensional reconstruction point-spread function and provide examples of superresolution for rotationally symmetric and angularly dependent objects. Finally, we investigate the role of the local quantum fluctuations in the superresolution process and evaluate the local noise figure for the reconstruction process.
A new scheme, superresolving Fourier-microscopy is proposed, which detects the spatial Fourier spectrum of the image created by an additional Fourier lens. This procedure amounts in directly imaging on the observation plane the field coming through the pupil, and all the field transmitted by the pupil is again concentrated on a finite region of space. Using decomposition on the basis of the prolate spheroidal functions, object reconstruction is shown to be in principle possible. The gain in superresolution achieved in this scheme over the classical Rayleigh limit is investigated numerically.
We present the quantum theory of superresolution for discrete subwavelength structures. It allows to formulate, in particular, the standard quantum limit of superresolution achieved for illumination of the structure by light in coherent state. Our theory is based on discrete prolate spheroidal sequences and functions which are the proper basis set of the problem. We demonstrate that the superresolution factor is much higher for discrete structures than for continuous objects for the same signal-to-noise ratio. This result is a clear illustration of the crucial role of a priori information in superresolution problems.
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 investigate analytically and numerically the role of quantum fluctuations in reconstruction of optical objects from diffraction-limited images. Taking as example of an input object two closely spaced Gaussian peaks, we demonstrate that one can improve the resolution in the reconstructed object over the classical Rayleigh limit. We show that the ultimate quantum limit of resolution in such a reconstruction procedure is determined not by diffraction but by the signal-to-noise ratio in the input object. We formulate a quantitative measure of super-resolution in terms of the optical point-spread function of the system.
Superresolution refers to the estimation of parameters of an image with an accuracy beyond standard classical techniques such as direct detection. In seminal work by Lu et al., a measurement to estimate the separation distance of two point sources (with a known centroid) was shown to achieve the quantum Cramer-Rao bound. This work made implicit use of reflection symmetry of the sources. Here we present a framework that uses more general symmetry in a constellation to construct a quantum measurement that achieves the quantum Cramer-Rao bound in estimation of parameters. We show how this technique can be used to estimate parameters simultaneously in symmetric point-source constellations with more than two point sources. In order to use symmetry explicitly, we make use discrete point spread functions in momentum space that maintain this symmetry. This framework allows us to use techniques from quantum computing such as Fourier transforms and linear optical circuits to implement the optimal measurement. To our knowledge, this is first work that shows for more than two point sources achievable quantum limits of estimation and modal transformations.
We discuss the ultimate limit imposed by quantum fluctuations of light for resolution of fine details in optical images. For this purpose, we extend in the quantum domain the classical analysis of the object reconstruction, or superresolution, in terms of prolate spheroidal function basis. We derive the expression for ultimate resolution limit in the reconstructed object using an illumination of the full object plane by a multimode squeezed vacuum. We show that the gain in resolution using multimode squeezed light is maximum when the Shannon number of the imaging system is close to unity.