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arXiv:2511.06098

Super-resolution with Fourier measurements

S. A. Wadood, Shaurya Aarav, Kevin Liang, Jason W Fleischer

physics.opticsquant-phSubmitted Nov 8, 2025• Announced Nov 11, 2025• 0 citations• Revised Dec 12, 2025 (v2)
v1 · Nov 8, 2025v2 · Dec 12, 2025
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Abstract

Resolving sources beyond the diffraction limit is important in imaging, communications, and metrology. Current image-based methods of super-resolution require phase information (either of the source points or an added filter) and perfect alignment with the centroid of the object. Both inhibit the practical application of these methods, as uniform motion and/or relative jitter destroy their assumptions. Here, we show that measuring intensity in the Fourier plane enables super-resolution without any of the issues of image-based methods. We start with the shift-invariance of the Fourier transform and the observation that the two-point position problem ${x_1,x_2}$ in the near field corresponds to the single-point wavenumber problem $k\ =2π/(x_2-x_1)$ in the far field. We consider the full range of mutual coherence and show that for fully coherent sources, the Fourier method saturates the quantum limit, i.e. it gives the best possible measurement. Similar results hold for sub-Rayleigh constellations of $N$ sources, which can act collectively as a spatially averaged metasurface and/or individually as elements of a phased-array antenna. The theory paves the way to merge Fourier optics with super-resolution techniques, enabling experimental devices that are both simpler and more robust than previous designs.

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Resolving sources beyond the diffraction limit is important in imaging, communications, and metrology. Current image-based methods of super-resolution require phase information (either of the source points or an added filter) and perfect alignment with the centroid of the object. Both inhibit the practical application of these methods, as uniform motion and/or relative jitter destroy their assumptions. Here, we show that measuring intensity in the Fourier plane enables super-resolution without any of the issues of image-based methods. We start with the shift-invariance of the Fourier transform and the observation that the two-point position problem x1,x2{x_1,x_2}x1​,x2​ in the near field corresponds to the single-point wavenumber problem k =2π/(x2−x1)k\ =2π/(x_2-x_1)k =2π/(x2​−x1​) in the far field. We consider the full range of mutual coherence and show that for fully coherent sources, the Fourier method saturates the quantum limit, i.e. it gives the best possible measurement. Similar results hold for sub-Rayleigh constellations of NNN sources, which can act collectively as a spatially averaged metasurface and/or individually as elements of a phased-array antenna. The theory paves the way to merge Fourier optics with super-resolution techniques, enabling experimental devices that are both simpler and more robust than previous designs.
The diffraction, or Rayleigh, limit was long thought to determine the ultimate resolving power of optical imaging devices. Using techniques from quantum statistical inference, Tsang et al. [1] demonstrated that this limit can be circumvented, and derived an alternative measurement strategy capable of resolving two incoherent point sources no matter how arbitrarily small their separation is. However, these measurements require perfect knowledge of the sources centroid position, and are extremely sensitive to any source of misalignment about this position [2] , [3] . In this work, we study the problem of super-resolution from a more global point of view.
The method of optical centroid measurement (OCM) has shown to exhibit spatial super-resolution with enhancements at the Heisenberg limit in plane wave interference experiments. In this work, the OCM method is for the first time used in an imaging setting where actual object features are observed. The OCM result is rederived in a near-field imaging formalism for a general imaging system and in full 2-D treatment. Analogies to coherent and incoherent imaging are shown Moreover, coherent OCM imaging is experimentally implemented for photon number N = 2, where an experimental setup is presented which allows to generate the desired entangled two-photon state containing the super-resolved image. This state is then imaged by an imaging system with a finite resolution defined by its point spread function (PSF). The centroid measurement of the two-photon states delivers then an image with a width of the PSF reduced by a factor 2 corresponding to the Heisenberg limit. In the experiment, the object is illuminated by a continuous-wave pump laser centred at 405nm with an output power of 30mW. A 4-f lens system images the object to the state preparation output plane. In the far-field plane between the lenses, a 5mm long periodically poled KTiOPO4 non-linear crystal generates photon pairs at 810nm by type-0, frequency-degenerate and collinear spontaneous parametric down-conversion. This generated OCM state is then imaged by a single lens and detected in coincidence by a fully digital 2-D sensor array with single-photon sensitivity and per-pixel sub-nanosecond time resolution. The OCM state is spectrally filtered at 810nm. Its imaging capability is compared to classical light sources including spatially coherent, monochromatic illumination at 405nm and 810 nm, as well as spatially incoherent light at 810 nm. The former is implemented using collimated lasers, the latter is a thermal light source spectrally filtered at 810 nm. The PSF of the different light sources are compared at low numerical aperture by imaging a focal point of 25μm Gaussian waist radius.
The resolution of imaging devices is ultimately limited by the diffraction of light. This limit can be circumvented by exploiting near-field phenomena, non-linearities of the sample or specific illumination schemes, which led to the recent explosion of optical super-resolution techniques. However, it would be desirable to reach the nanoscale in the most general way possible, with far-field linear imaging of uniformly illuminated samples. In the last few years, the answer to that came with the field of quantum-inspired super-resolution (QISR), where quantum metrological techniques were used to find novel measurement schemes that, beyond the conventional direct imaging of light, would achieve higher precisions. Interestingly, these ideas would always in some way or another involve transverse spatial modes of light. This thesis presents our work on the subject, with novel theoretical and experimental results. Firstly, we develop the traditional theory of QISR, with has mostly focused on either the estimation of separation of two point sources or on geometric moments of samples. Then, we introduce our own results on optical imaging of general distributions of point sources, in a step to take QISR from toy examples to real imaging scenarios. Secondly, we apply all the concepts studied on a experimental, proof-of-principle implementation of a QISR scheme dubbed Hermite-Gaussian Microscopy (HGM), which uses heterodyne detection with several Hermite-Gaussian modes as local oscillator in order to reconstruct the object sample and achieve super-resolution. We report the techniques necessary for the manipulation of spatial modes of light, as well as our results, where we demonstrated 2D imaging with twofold resolution enhancement beyond the diffraction limit.
Optical precision measurement is evolving from conventional intensity and phase readout toward a paradigm centered on mode selection, statistical estimation, and information extraction. Conventional intensity- and phase-based schemes can achieve high sensitivity, but their performance is often constrained by finite-aperture diffraction, reference stability, optical path control, and environmental perturbations in systems. Quantum estimation theory provides a physical framework for optical resolution and measurement precision by optimizing the measurement basis. Inspired by spatial mode demultiplexing, this paper extends quantum parameter estimation from incoherent point-source resolution to small-displacement measurement. Metasurface enabled mode control converts displacement induced optical-field variations into measurable changes in orthogonal modal coefficients, providing a route for extracting sub-diffraction-scale displacement information. This paper introduces Rayleigh’s curse, Fisher information, and spatial mode projection, discusses their applications in one-dimensional displacement measurement and two-dimensional super-resolution imaging, and outlines future directions in digital optics, Engineering Optics 2.0, mode-basis optimization, two-dimensional information reconstruction, and intelligent optical measurement systems for engineering applications.
Long baseline diffraction-limited optical aperture synthesis technology by interferometry plays an important role in scientific study and practical application. In contrast to amplitude (phase) interferometry, intensity interferometry -- which exploits the quantum nature of light to measure the photon bunching effect in thermal light -- is robust against atmospheric turbulence and optical defects. However, a thermal light source typically has a significant divergence angle and a low average photon number per mode, forestalling the applicability over long ranges. Here, we propose and demonstrate active intensity interferometry for super-resolution imaging over the kilometer range. Our scheme exploits phase-independent multiple laser emitters to produce the thermal illumination and uses an elaborate computational algorithm to reconstruct the image. In outdoor environments, we image two-dimension millimeter-level targets over 1.36 kilometers at a resolution of 14 times the diffraction limit of a single telescope. High-resolution optical imaging and sensing are anticipated by applying long-baseline active intensity interferometry in general branches of physics and metrology.
For more than a century, the diffraction limit has defined the resolution achievable by passive optical imaging systems. Although some resolution improvement can be gained through classical data processing of the image, it is limited by the noise arising from quantum nature of light. Minimizing the effect of this noise requires quantum treatment of optical imaging. By reformulating imaging as a problem of quantum measurement and estimation, it becomes possible to identify optimal detection strategies that recover spatial information previously thought inaccessible. This review summarizes the theoretical framework that underpins this development, from the formulation of quantum Cramer-Rao bounds and Chernoff bounds to the construction of receivers that attain them, such as those based on spatial-mode demultiplexing. We show how these methods can beat conventional imaging in the classification, localization, and imaging of sub-Rayleigh incoherent sources. We then discuss extensions to multiparameter and partially coherent scenarios, and highlight the unifying connections between estimation and discrimination tasks. Finally, we survey recent experimental demonstrations that approach quantum-limited resolution and outline emerging applications in microscopy, astronomy, and optical sensing.