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.Lord Rayleigh suggested in 1879 that two incoherent optical point sources should be separated by a diffraction-limited spot size for them to be resolved [1]. This criterion has since become the most influential measure of imaging resolution. Under the modern advent of rigorous statistics and image processing, Rayleigh's criterion remains a curse. When the image is noisy, necessarily so owing to the quantum nature of light [2], and Rayleigh's criterion is violated, it becomes much more difficult to estimate the separation accurately by conventional imaging methods [3][4][5]. Modern superresolution techniques in microscopy [6][7][8] can circumvent Rayleigh's criterion by making sources radiate in isolation, but such techniques require careful control of the fluorescent emissions, making them difficult to use for microscopy and irrelevant to astronomy.Here we show that, contrary to conventional wisdom, the separation between two incoherent optical sources can be estimated accurately via linear optics and photon counting (LOPC) even if Rayleigh's criterion is severely violated. Our theoretical model here is based on the semiclassical theory of photodetection with Poissonian noise, which is a widely accepted statistical model for lasers [2] as well as weak thermal [9,10] or fluorescent [5,11] light in astronomy and microscopy. The semiclassical model is consistent with the quantum model proposed in Ref. [12] for weak incoherent sources and the mathematical formalisms are similar, but the semiclassical model has the advantage of being applicable also to lasers, which are important sources for remote-sensing, testing, and proof-of-concept experiments. The semiclassical theory also avoids a quantum description of light and offers a more pedagogical perspective. Compared with the full semiclassical theory in Ref. [13], the Poissonian model is invalid for strong thermal sources but more analytically tractable.Consider J optical modes and a column vector of complex field amplitudes α = (α 1 , . . . , α J ) ⊤ within one coherence time interval. The amplitudes are normalized such that |α j | 2 is equal to the energy in each mode in units of quanta. The central quantity in statistical optics is the mutual coherence matrix [2,9]

Ever since the inception of gravitational-wave detectors, limits imposed by quantum mechanics to the detection of time-varying signals have been a subject of intense research and debate. Drawing insights from quantum information theory, quantum detection theory, and quantum measurement theory, here we prove lower error bounds for waveform detection via a quantum system, settling the long-standing problem. In the case of optomechanical force detection, we derive analytic expressions for the bounds in some cases of interest and discuss how the limits can be approached using quantum control techniques.Comment: v1: first draft, 5 pages; v2: updated and extended, 5 pages + appendices, 2 figures; v3: 8 pages and 3 figure

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.PACS numbers: 42.30.-d, 42.50.-p, 06.20.-f The Rayleigh criterion for resolving two incoherent optical point sources [1] is the most widely used benchmark for the resolving power of an imaging system. According to it, the sources can be resolved by direct imaging only if they are separated by at least the diffraction-limited spot size of the point-spread function of the imaging system. While the criterion is heuristic and does not take into account the intensity of the sources or the measurement shot noise, recent work [2-5] has made it rigorous by taking as resolution measure the classical Cramér-Rao lower bound (CRB) of estimation theory [6] on the mean squared error (MSE) of any unbiased estimate of the separation of the sources using spatially-resolved imageplane photon counting. These works showed that if the detected average photon number per mode N s ≪ 1, the MSE of any unbiased estimator based on direct imaging diverges as the source separation decreases to zero over an interval comparable to the Rayleigh limit. This phenomenon, dubbed Rayleigh's curse in [7], stems from the indistinguishability between the photons coming from the two sources and imposes a fundamental limitation of direct imaging in resolving sources much closer than the spot size, even when the measured photon number is taken into account. Recent developments in far-field microscopy [8] sidestep Rayleigh's curse by preventing multiple sources from emitting simultaneously, but control over the emission properties of sources is unavailable in target sensing or astronomical imaging.While the development of novel quantum states of light and measurement techniques has given rise to the vast field of quantum imaging [9], fundamental quantum limits in resolving two incoherent sources have been largely neglected since the early days of quantum estimation theory [10,11]. Recently, the coherent [12] and incoherent [7] two-source resolution problems were revisited using the quantum Cramér-Rao bound (QCRB) [11,13] that accounts for all (unbiased) measurement techniques allowed by quantum mechanics. Under a weak-source assumption similar to that in [2][3][4][5], it was found in [7] that the QCRB showed no depe...

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 photon number-resolving and on-off detectors. The analysis is supported by Monte-Carlo simulations of the maximum likelihood estimator for the source separation, demonstrating the superlocalization effect for separations well below that set by the Rayleigh criterion. Simulations indicating the robustness of SLIVER to mismatch between the optical axis and the centroid are also presented. The results are valid for any imaging system with a circularly symmetric point-spread function.

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