We consider the problem of estimating the spatial separation between two mutually incoherent point light sources using the super-resolution imaging technique based on spatial mode demultiplexing (SPADE) with noisy detectors. We show that in the presence of noise, the resolution of the measurement is limited by the signal-to-noise ratio (SNR) and the minimum resolvable spatial separation has a characteristic dependence of [Formula: see text]. Several detection techniques, including direct photon counting, as well as homodyne and heterodyne detection are considered.
We experimentally demonstrate a universal, parameter-independent test for asymmetric source discrimination below the diffraction limit. The test exploits spatial-mode demultiplexing (SPADE) and single-photon detection to discriminate the presence of a faint source in the vicinity of a brighter one. We develop a complete theory of the test in the presence of arbitrary modal crosstalk and show that the crosstalk-affected test retains a favorable error exponent over diffraction-limited direct imaging, provided that the crosstalk remains below a threshold
Cmax≃1/9. This establishes a quantitative condition under which the subdiffraction advantage of SPADE survives realistic imperfections. We experimentally test the scheme for a range of source separations and intensity ratios, without requiring prior knowledge of these parameters. The measured false-negative error rates are in good agreement with the theoretical model and systematically outperform direct imaging. These results demonstrate that SPADE provides a robust and experimentally accessible advantage for asymmetric subdiffraction hypothesis testing in the photon-starved regime, even in the presence of realistic modal crosstalk.
Spatial-mode demultiplexing (SPADE) has recently been adopted to measure the separation in the transverse plane between two incoherent point-like sources with sub-wavelength separation. It has been argued that this approach may yield extraordinary performances in the photon-counting regime. Here, we explore SPADE as a tool for precision measurements in the regime of bright, incoherent sources. First we analyze the general problem of estimating the second moments of the source's intensity distribution, for an extended incoherent source of any shape. Our theory predicts a substantial improvement in signal-to-noise ratio (SNR) of SPADE over direct imaging in the sub-wavelength regime. Second, we present an experimental application of SPADE to the case of two point-like, bright sources. We demonstrate the use of this setup for the estimation of the transverse separation and for the estimation of the relative intensity, confirming the expected improvement in SNR.
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.
High-resolution array detectors are widely used in single-particle tracking, but their performance is limited by excess noise from background light and dark current. As pixel resolution increases, the diminished signal per pixel exacerbates susceptibility to noise, degrading tracking accuracy. To overcome this limitation, we use spatial-mode demultiplexing (SPADE) as a noise-robust approach for estimating the motion characteristics of an optical pointlike source. We show that SPADE efficiently concentrate the information into a few key spatial modes, drastically reducing the number of detectors while maintaining high estimation precision. Furthermore, we enhance the robustness of the estimation against excess noise by elaborately designing the modes to be decomposed. We demonstrate, both theoretically and experimentally, that a SPADE with two specific modes outperforms direct imaging in estimating the microoscillation frequency of an optical point source in the presence of excess noise.
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.