Conventional incoherent imaging based on measuring the spatial intensity distribution in the image plane faces the resolution hurdle described by the Rayleigh diffraction criterion. Here, we demonstrate theoretically using the concept of the Fisher information that quadrature statistics measured by means of array homodyne detection enables estimation of the distance between two incoherent point sources well below the Rayleigh limit for sufficiently high signal-to-noise ratio. This capability is attributed to the availability of spatial coherence information between individual detector pixels acquired using the coherent detection technique. A simple analytical approximation for the precision attainable in the sub-Rayleigh region is presented. Furthermore, an estimation algorithm is proposed and applied to Monte Carlo simulated data.
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. W…
Rayleigh's criterion states that it becomes essentially difficult to resolve two incoherent optical point sources separated by a distance below the width of point spread functions (PSF), namely in the subdiffraction limit. Recently, researchers have achieved superresolution for two incoherent point sources with equal strengths using a new type of measurement technique, surpassing Rayleigh's criterion. However, situations where more than two point sources needed to be resolved have not been fully investigated. Here we prove that for any incoherent sources with arbitrary strengths, a one- or two…
The Rayleigh limit has so far applied to all microscopy techniques that rely on linear optical interaction and detection in the far field. Here we demonstrate that detecting the light emitted by an object in higher-order transverse electromagnetic modes (TEMs) can help achieving sub-Rayleigh precision for a variety of microscopy-related tasks. Using optical heterodyne detection in TEM01, we measure the position of coherently and incoherently emitting objects to within 0.0015 and 0.012 of the Rayleigh limit, respectively, and determine the distance between two incoherently emitting slits positi…
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 farfield 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, measuremen…
We obtain the multiple-parameter quantum Cramér-Rao bound for estimating the transverse Cartesian components of the centroid and separation of two incoherent optical point sources using an imaging system with finite spatial bandwidth. Under quite general and realistic assumptions on the point-spread function of the imaging system, and for weak source strengths, we show that the Cramér-Rao bounds for the $x$ and $y$ components of the separation are independent of the values of those components, which may be well below the conventional Rayleigh resolution limit. We also propose two linear optics…
Using a semiclassical model of photodetection with Poissonian noise and insights from quantum metrology, we prove that linear optics and photon counting can optimally estimate the separation between two incoherent point sources without regard to Rayleigh's criterion. The model is applicable to weak thermal or fluorescent sources as well as lasers.
We analyze the fundamental resolution of incoherent optical point sources from the perspective of a quantum detection problem: deciding whether the optical field on the image plane is generated by one source or two weaker sources with arbitrary separation. We investigate the detection performances of two measurement methods recently proposed by us to enhance the estimation of the separation. For the detection problem, we show that the method of binary spatial-mode demultiplexing is quantum-optimal for all values of separations, while the method of image-inversion interferometry is near-optimal…
It has been argued that, for a spatially invariant imaging system, the information one can gain about the separation of two incoherent point sources decays quadratically to zero with decreasing separation, an effect termed Rayleigh's curse. Contrary to this belief, we identify a class of point spread functions with a linear information decrease. Moreover, we show that any well-behaved symmetric point spread function can be converted into such a form with a simple nonabsorbing signum filter. We experimentally demonstrate significant superresolution capabilities based on this idea.
A novel interferometric method - SLIVER (Super Localization by Image inVERsion interferometry) - is proposed for estimating the separation of two incoherent point sources with a mean squared error that does not deteriorate as the sources are brought closer. The essential component of the interferometer is an image inversion device that inverts the field in the transverse plane about the optical axis, assumed to pass through the centroid of the sources. The performance of the device is analyzed using the Cramér-Rao bound applied to the statistics of spatially-unresolved photon counting using ph…
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 p…
No imaging apparatus can produce perfect images: spatial resolution is limited by the Rayleigh diffraction bound that is a consequence of the imager's finite spatial extent. We show some N-photon strategies that permit resolution of details that are smaller than this bound, attaining either a 1/sqrt(N) enhancement (standard quantum limit) or a 1/N enhancement (Heisenberg limit) over standard techniques. In the incoherent imaging regime, the methods presented are loss resistant, because they can be implemented with classical-state light sources. Our results may be of importance in many applicat…
We propose and demonstrate a photon-efficient optical classifier to overcome the Rayleigh limit in spatial resolution. It utilizes mode-selective sum frequency generation and single-pixel photon detection to resolve closely spaced incoherent sources based on photon counting statistics. Super-resolving and photon efficient, this technique can find applications in microscopy, light detection and ranging (LiDAR), and astrophysics.
We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using alinear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source 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 …
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…
We analyze the fundamental quantum limit of the resolution of an optical imaging system from the perspective of the detection problem of deciding whether the optical field in the image plane is generated by one incoherent on-axis source with brightness $ε$ or by two $ε/2$-brightness incoherent sources that are symmetrically disposed about the optical axis. Using the exact thermal-state model of the field, we derive the quantum Chernoff bound for the detection problem, which specifies the optimum rate of decay of the error probability with increasing number of collected photons that is allowed …
Recent work considered the ultimate (quantum) limit of the precision of estimating the distance between two point objects. It was shown that the performance gap between the quantum limit and that of ideal continuum image-plane direct detection is the largest for highly sub-Rayleigh separation of the objects, and that a pre-detection mode sorting could attain the quantum limit. Here we extend this to a more general problem of estimating the length of an incoherently radiating extended (line) object. We find, as expected by the Rayleigh criterion, the Fisher information (FI) per integrated photo…
In this work we investigate the quantum information theoretical limits to several tasks related to lifetime estimation and discrimination of a two-level spontaneous optical emitter. We focus in particular on the model problem of resolving two mutually incoherent exponential decays with highly overlapping temporal probability profiles. Mirroring recent work on quantum-inspired super-resolution of point emitters, we find that direct lifetime measurement suffers from an analogue of "Rayleigh's Curse" when the time constants of the two decay channels approach one another. We propose alternative me…
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 sub-wavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the po…
The wave-particle duality of light introduces two fundamental problems to imaging, namely, the diffraction limit and the photon shot noise. Quantum information theory can tackle them both in one holistic formalism: model the light as a quantum object, consider any quantum measurement, and pick the one that gives the best statistics. While Helstrom pioneered the theory half a century ago and first applied it to incoherent imaging, it was not until recently that the approach offered a genuine surprise on the age-old topic by predicting a new class of superior imaging methods. For the resolution …
Passive optical superresolution at the quantum limit
University of Oxford（牛津大学）；RTX BBN Technologies（RTX BBN技术公司）；University of Maryland, College Park（马里兰大学帕克分校）；National University of Singapore（新加坡国立大学）；University of Nottingham（诺丁汉大学）；Sorbonne Université（索邦大学）
A recently identified class of receivers which demultiplex an optical field into a set of orthogonal spatial modes prior to detection can surpass canonical diffraction limits on spatial resolution for simple incoherent imaging tasks. However, these mode-sorting receivers tend to exhibit high sensitivity to contextual nuisance parameters (e.g., the centroid of a clustered or extended object), raising questions on their viability in realistic imaging scenarios where little or no prior information about the scene is available. We propose a multi-stage passive imaging strategy which segments the t…
Estimating the angular separation between two incoherently radiating monochromatic point sources is a canonical toy problem to quantify spatial resolution in imaging. In recent work, Tsang {\em et al.} showed, using a Fisher Information analysis, that Rayleigh's resolution limit is just an artifact of the conventional wisdom of intensity measurement in the image plane. They showed that the optimal sensitivity of estimating the angle is only a function of the total photons collected during the camera's integration time but entirely independent of the angular separation itself no matter how smal…
In this paper we study the resolution of images illuminated by sources composed of $N+1$ photons in which one non-degenerate photon is entangled with $N$ degenerate photons. The $N$ degenerate photons illuminate an object and are collected by an $N$ photon detector. The signal from the $N$ photon detector is measured in coincidence with the non-degenerate photon giving rise to a ghost image. We discuss the case of three photons in various configurations and generalize to $N+1$. Using the Rayleigh criterion, we find that the system may give an improvement in resolution by a factor of $N$ compar…
