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arXiv:1411.2954 · Optica

Quantum limits to optical point-source localization

Mankei Tsang

Published Jul 10, 2015• 70 citations
arXivHTMLPDF

Abstract

Motivated by the importance of optical microscopes to science and engineering, scientists have pondered for centuries how to improve their resolution and the existence of fundamental resolution limits. In recent years, a new class of microscopes that overcome a long-held belief about the resolution have revolutionized biological imaging. Termed "superresolution" microscopy, these techniques work by accurately locating optical point sources from far field. To investigate the fundamental localization limits, here I derive quantum lower bounds on the error of locating point sources in free space, taking full account of the quantum, nonparaxial, and vectoral nature of photons. These bounds are valid for any measurement technique, as long as it obeys quantum mechanics, and serve as general no-go theorems for the resolution of microscopes. To arrive at analytic results, I focus mainly on the cases of one and two classical monochromatic sources with an initial vacuum optical state. For one source, a lower bound on the root-mean-square position estimation error is on the order of $λ_0/\sqrt{N}$, where $λ_0$ is the free-space wavelength and $N$ is the average number of radiated photons. For two sources, owing to the statistical effect of nuisance parameters, the error bound diverges when their radiated fields overlap significantly. The use of squeezed light to enhance further the accuracy of locating one classical point source and the localization limits for partially coherent sources and single-photon sources are also discussed. The presented theory establishes a rigorous quantum statiscal inference framework for the study of superresolution microscopy and points to the possibility of using quantum techniques for true resolution enhancement.

DOI
10.1364/optica.2.000646

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For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.
In recent years quantum metrology and quantum sensing have enabled the advancement of quantum technologies and research in fundamental physics. Through statistical analysis, in the context of parameter estimation, the amount of information about a parameter of interest encoded in a quantum state can be quantified. In this thesis we focus on the application of quantum estimation theory to imaging and spectroscopy with quantum light. We study the fundamental bound of the mean square error for an unbiased estimator in terms of the quantum Fisher information (QFI) for two problems, one in microscopy and one in spectroscopy. Firstly, we study the problem of localising multiple point sources below the diffraction limit. We show that localisation microscopy of multiple weak, incoherent point sources with possibly different intensities in one spatial dimension is equivalent to estimating the amplitudes of a classical mixture of coherent states of a simple harmonic oscillator. We obtain the QFI matrix elements analytically. In the regime of arbitrarily small separations we find it to be no more than rank two – implying that no more than two independent parameters can be estimated irrespective of the number of point sources. We use the eigenvalues of the classical and quantum Fisher information matrices to compare the performance of spatial-mode demultiplexing and direct imaging in localisation microscopy with respect to the quantum limits. Secondly, we study the estimation of the electric dipole moment (EDM) of a two-level atom through its interaction with quantum pulses of light in free space. We derive analytical expressions for the states of one-photon wavepackets and entangled photon pairs (EPP). We numerically calculate the QFI for different states of light, including coherent and squeezed pulses, and compare their performance in estimating the EDM. We find that the one-photon wavepacket and the EPP have similar performances, while the performance of the entangled photon pair is not improved by increasing the entanglement of the EPP. Our results indicate that using Fock states to estimate the EDM of a two-level atom is preferable to using entangled light.
Rayleigh's criterion for resolving two incoherent point sources has been the most influential measure of optical imaging resolution for over a century.In the context of statistical image processing, violation of the criterion is especially detrimental to the estimation of the separation between the sources, and modern far-field superresolution techniques rely on suppressing the emission of close sources to enhance the localization precision.Using quantum optics, quantum metrology, and statistical analysis, here we show that, even if two close incoherent sources emit simultaneously, measurements with linear optics and photon counting can estimate their separation from the far field almost as precisely as conventional methods do for isolated sources, rendering Rayleigh's criterion irrelevant to the problem.Our results demonstrate that superresolution can be achieved not only for fluorophores but also for stars.
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Point source localization is a problem of persistent interest in optical imaging. In particular, a number of widely used biological microscopy techniques rely on precise three-dimensional localization of single fluorophores. As emitter depth localization is more challenging than lateral localization, considerable effort has been spent on engineering the response of the microscope in a way that reveals increased depth information. Here, we prove the (sub)optimality of these approaches by deriving and comparing to the measurement-independent quantum Cramér-Rao bound (QCRB). We show that existing methods for depth localization with single-objective collection exceed the QCRB, and we gain insight into the bound by proposing an interferometer arrangement that approaches it. We also show that for light collection with two opposed objectives, an established interferometric technique globally reaches the QCRB in all three dimensions simultaneously, and so this represents an interesting case study from the point of view of quantum multiparameter estimation.
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.