Interferometric methods have been recently investigated to achieve sub-Rayleigh imaging and precision measurements of faint incoherent sources up to the ultimate quantum limit. Here we consider single-photon imaging of two point-like emitters of unequal intensity. This is motivated by the fact that pairs of natural emitters typically have unequal brightness, for example, binary star systems and exoplanets. We address the problem of estimating the transverse separation d or the relative intensity . Our theoretical analysis shows that the associated statistical errors are qualitatively different from the case of equal intensity. We employ multi-plane light conversion technology to implement Hermite–Gaussian (HG) spatial-mode demultiplexing (SPADE), and demonstrate sub-Rayleigh measurement of two emitters with a Gaussian point-spread function. The experimental errors are comparable with the theoretical bounds. The latter are benchmarked against direct imaging, yielding an improvement in the signal-to-noise ratio, which may be significant when the primary source is much brighter than the secondary one, for example, as for imaging of exoplanets.
We propose a Bayesian evidence-based inference framework based on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom measurement, SPADE require no collective detection and its optimal for asymptotically large samples. Our method avoids ad hoc statistical constructs and relies solely on the information contained in the data, with all assumptions entering only through the likelihood model and prior beliefs. Using experimental evidence, we demonstrate the superior resolving performance of SPADE over direct imaging from a new and extensible perspective; one that naturally generalizes to multiple sources and offers a practical robust approach to analyzing quantum-enhanced superresolution.
In optical imaging, diffraction strongly degrades the performance of conventional intensity-based estimation once the separation between two sources is below the Rayleigh-Abbe limit. Recent developments showed that this limitation can be surpassed using spatial-mode demultiplexing (SPADE), and this was demonstrated in several experiments for two equal-intensity spots of unknown separation. When there are multiple unknown parameters, as in the case of several unequal sources with unknown intensities and unknown separation, cross-talk among the parameters makes the multi-parameter estimation problem significantly more challenging. In this paper, we adapt super-resolved position localization by inversion of coherence along an edge (SPLICE) to estimate both the separation and relative intensity of two incoherent sources simultaneously. We demonstrate a clear advantage over direct imaging (DI), achieving a root mean squared error (RMSE) approximately
50% larger than the quantum limit and a sixfold improvement over DI within the range of parameters we tested. This improvement can be even greater for smaller separations and larger intensity imbalances when crosstalk is suppressed.
Recent works employing tools from quantum optics and quantum metrology proposed a new passive imaging technique that allows to resolve details far below the diffraction limit. This technique is based on replacing standard spatially-resolved intensity measurements, e.g. at each pixel of a camera, with spatial-mode demultiplexing (SpaDe) measurements that allow to acquire information in a more efficient way. In this contribution, we want to provide an intuitive explanation of why such a SpaDe approach is so effective, and illustrate how we used these idea to discriminate one point source from two, and to estimate the separation between two incoherent sources.
We study far-field discrimination between one and two incoherent point sources in the singular regime of weak and closely spaced emitters. Under ideal alignment, spatial-mode demultiplexing (SPADE) attains the quantum-optimal large-sample Stein exponent, but the finite-photon behavior near the one-source boundary and the effect of realistic imperfections remain less understood. Using singular learning theory, we analyze both the aligned and misaligned problems. In the aligned Gaussian case, for prior densities that are smooth and strictly positive in the physical
(ε,s) coordinates near the singularity of the aligned model, direct imaging and SPADE share the same real log canonical threshold
λ=1/2 but have different multiplicities, yielding distinct Bayes free-energy asymptotics. A fixed nonzero misalignment removes the exact support mismatch of ideal SPADE: locally, the fixed-offset binary-SPADE Kullback--Leibler function has the normal-crossing form
K≍ε2s2, giving
(λ,m)=(1/2,2) under the same prior class, as for direct imaging. The local separation scale nevertheless depends on the source-position convention. Moreover, full Hermite--Gaussian mode counting about an offset sorter axis has a reflection-induced KL-zero branch at
s∗=2θ, where the two-source alternative becomes indistinguishable from the null. Pointwise finite-
n binary-SPADE calculations exhibit the corresponding power collapse. These results identify measurement support, nuisance geometry, prior weighting, and identifiability as structural ingredients that must be tracked in finite-photon quantum discrimination.
I propose classical and quantum limits to the statistical resolution of two incoherent optical point sources from the perspective of minimax parameter estimation. Unlike earlier results based on the Cramér-Rao bound (CRB), the limits proposed here, based on the worst-case error criterion and a Bayesian version of the CRB, are valid for any biased or unbiased estimator and obey photon-number scalings that are consistent with the behaviours of actual estimators. These results prove that, from the minimax perspective, the spatial-mode demultiplexing measurement scheme recently proposed by Tsang, Nair, and Lu [Phys. Rev. X 2016, 6 031033.] remains superior to direct imaging for sufficiently high photon numbers.