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Spatial mode demultiplexing imaging in single photon regime

Luigi Santamaria Amato, Fabrizio Sgobba, Cosmo Lupo

Published May 30, 2025• 0 citations
Publisher

Abstract

The resolution of direct imaging, which is the most widespread technique for remote observation in the optical domain, is fundamentally limited by diffraction. It has been recently shown that the diffraction limit can be overcome, even with passive optics, using interferometric far-field techniques like spatial mode demultiplexing (SPADE). We present our results on sub-diffraction imaging and spectroscopy of a pair of incoherent point-like emitters using SPADE in the single-photon regime. A theoretical analysis provides a comparison with direct imaging and explains the improvement in terms of signal-to-noise ratio. The application of single-photon SPADE for detection of more complex objects beyond a pair of point-emitters is underway for deployment in earth observation and object detection. The results are presented in the framework of the Quantum Imaging for Exoplanet detection (QUEXO) project.

DOI
10.1117/12.3053582

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The resolution of direct imaging, which is the most widespread technique for remote observation in the optical domain, is fundamentally limited by diffraction. It has been recently shown that the diffraction limit can be overcome, even with passive optics, using interferometric far-field techniques like spatial mode demultiplexing (SPADE). We present our results on sub-diffraction imaging and spectroscopy of a pair of incoherent point-like emitters using SPADE in the single-photon regime. A theoretical analysis provides a comparison with direct imaging and explains the improvement in terms of signal-to-noise ratio. The application of single-photon SPADE for detection of more complex objects beyond a pair of point-emitters is underway for deployment in earth observation and object detection. The results are presented in the framework of the Quantum Imaging for Exoplanet detection (QUEXO) project.
The resolution of direct imaging, the most common method for remote observation in the optical domain, is inherently constrained by diffraction. Recent studies have demonstrated that this diffraction limit can be surpassed, even with passive optics, through interferometric far-field techniques such as spatial mode demultiplexing (SPADE) or super localization by image inversion interferometry (SLIVER). We share our findings on sub-diffraction imaging of two incoherent point-like sources using SPADE and SLIVER.
I propose a spatial-mode demultiplexing (SPADE) measurement scheme for the far-field imaging of spatially incoherent optical sources. For any object too small to be resolved by direct imaging under the diffraction limit, I show that SPADE can estimate its second or higher moments much more precisely than direct imaging can fundamentally do in the presence of photon shot noise. I also prove that SPADE can approach the optimal precision allowed by quantum mechanics in estimating the location and scale parameters of a subdiffraction object. Realizable with far-field linear optics and photon counting, SPADE is expected to find applications in both fluorescence microscopy and astronomy.
I present a semiclassical analysis of a spatial-mode demultiplexing (SPADE) measurement scheme for far-field incoherent optical imaging under the effects of diffraction and photon shot noise. Building on previous results that assume two point sources or the Gaussian point-spread function, I generalize SPADE for a larger class of point-spread functions and evaluate its errors in estimating the moments of an arbitrary subdiffraction object. Compared with the limits to direct imaging set by the Cram\'er-Rao bounds, the results show that SPADE can offer far superior accuracy in estimating second- and higher-order moments.
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.
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.
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.