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SPIE Quantum Technologies for Defence and Security

Parameter estimation approach to super resolution

Alexander Boeschoten, Antonin Grateau, Clémentine Rouvière, Ilya Karuseichyk, David Barral, Giacomo Sorelli, Mattia Walschaers, Manuel Gessner, Claude Fabre, Nicolas Treps

Published Nov 15, 2024• 0 citations
Publisher

Abstract

The resolution of optical systems, formulated as the smallest possible distance between two point sources for which they still can be dissolved, was for a long time believed to be limited by diffraction, formulated by the Rayleigh criterion. Recent advancements in quantum metrology have shown, by evaluation of the Quantum Cramér Rao bound (QCRB), that the Rayleigh criterion is not a fundamental limit. In our experiment, spatial mode demultiplexing (SPADE) is used to estimate the separation of the sources orders of magnitude below the Rayleigh limit. The experiment is extended to incorporate the measurement of additional parameters, such as power imbalance and centroid position of the two sources, bringing it closer to real-world applicability.

DOI
10.1117/12.3031253

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Due to the wave nature of light, optical imaging is inherently subject to diffraction, which typically dictates the resolution limit for two-point sources. A practical way to describe this phenomenon is through Rayleigh’s criterion, which states that two sources are barely resolvable when the minimum of one source's diffraction pattern coincides with the maximum of the other. However, recent advances in optical quantum metrology have shown that this classical limit can be surpassed by leveraging the spatial correlations within the light field. Specifically, decomposing the light into a Hermite-Gauss mode basis provides substantially more information than traditional direct imaging, which measures local intensity [1,2,3]. In this work, we implement a spatial demultiplexing technique to estimate the separation between light sources of unequal brightness. This unknown imbalance adds complexity to the estimation task, making it inherently a multiparameter problem and thus requiring a multimode approach [4]. By optimizing the measurement strategy and introducing a shift between two demultiplexers, we achieve sub-Rayleigh resolution in distance measurements without assuming prior knowledge of the intensity imbalance or alignment. [1] Tsang, M., Nair, R., & Lu, X. M. (2016). Quantum theory of superresolution for two incoherent optical point sources. Physical Review X, 6(3), 031033. [2] Sorelli, G., Gessner, M., Walschaers, M., & Treps, N. (2021). Optimal observables and estimators for practical superresolution imaging. Physical Review Letters, 127(12), 123604. [3] Rouvière, C., Barral, D., Grateau, A., Karuseichyk, I., Sorelli, G., Walschaers, M., & Treps, N. (2024). Ultra-sensitive separation estimation of optical sources. Optica, 11(2), 166-170. [4] Řeháček, J., Hradil, Z., Koutný, D., Grover, J., Krzic, A., & Sánchez-Soto, L. L. (2018). Optimal measurements for quantum spatial superresolution. Physical review A, 98(1), 012103.

SPIE Quantum Technologies for Defence and Security•Oct 29, 2025•0 citations
The resolution of optical systems, formulated as the smallest possible distance between two point sources for which they still can be dissolved, was for a long time believed to be limited by diffraction, formulated by the Rayleigh criterion. Recent advancements in quantum metrology have shown, by evaluation of the Quantum Cramér Rao bound (QCRB), that the Rayleigh criterion is not a fundamental limit. In our experiment, spatial mode demultiplexing (SPADE) is used to estimate the separation of the sources orders of magnitude below the Rayleigh limit. The experiment is extended to incorporate the measurement of additional parameters, such as power imbalance and centroid position of the two sources, bringing it closer to real-world applicability.
We experimentally demonstrate the simultaneous estimation of the three parameters characterizing a pair of incoherent optical sources in the sub-Rayleigh regime, enabling super-resolved scene characterization. Using spatial-mode demultiplexing (SPADE) with two demultiplexers--one deliberately shifted--we determine separations well below the diffraction limit and achieve sensitive joint estimation of separation, centroid, and relative brightness over a broad range of scene configurations in a single experimental setting. We benchmark our performance using Fisher-information-based Cramér-Rao bounds, and discuss the corresponding quantum limits. We investigate two complementary scenarios: a realistic case with slightly non-identical sources, and an idealized case of indistinguishable sources.
Abstract Achieving resolution in the sub-Rayleigh regime (superresolution) is one of the rapidly developing topics in quantum optics and metrology. Recently, it was shown that perfect measurement based on spatial mode demultiplexing (SPADE) in Hermite–Gauss modes allows one to reach the quantum limit of precision for estimation of separation between two weak incoherent stationary sources. Since then, different imperfections such as misalignment or crosstalk between modes have been studied to check how this result translates into more realistic experimental setups. In this paper, we consider another deviation from the perfect setup by discarding the assumption about the stationarity of the sources. This is relevant for example for astrophysical applications where planets necessarily orbit around the star. We analyze two examples of dynamics: rotations and oscillations, showing the robustness of the SPADE-based measurement against them. The analysis is based on Fisher information, which allows one to obtain the precision limit through Cramér–Rao bound. Furthermore, we formulate a measurement algorithm that allows for the reduction of one parameter for estimation (system orientation angle) in the stationary sources scenario, maintaining the measurement precision despite the lack of knowledge about this parameter.
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%50 \%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.
Resolving light sources below the diffraction limit is a fundamental task both for astronomy and microscopy. Several recent works analysed this problem through the lens of quantum parameter estimation theory and proved that the separation between two point sources can be estimated at the quantum limit using intensity measurements after spatial-mode demultiplexing. We implement this technique and provide an optimal estimator based on a linear combination of demultiplexed intensity measurements [1–3]. Our experimental setup allows for the generation of the images of two sources, with tunable mutual coherence, as well as for spatial mode demultiplexing to estimate their separation [4].
The development of super-resolution techniques, i.e., allowing for efficient resolution below the Rayleigh limit, became one of the important branches in contemporary optics and metrology. Recent findings show that perfect spatial mode demultiplexing (SPADE) into Hermite-Gauss modes followed by photon counting enables one to reach the quantum limit of precision in the task of estimation of separation between two weak stationary sources in the sub-Rayleigh regime. In order to check the limitations of the method, various imperfections such as misalignment or crosstalk between the modes were considered. Possible applications of the method in microscopy call for the adaptive measurement scheme, as the position of the measured system can evolve in time, causing non-negligible misalignment. In this paper, we examine the impact of Brownian motion of the center of the system of two weak incoherent sources on adaptive SPADE measurement precision limits. The analysis is carried out using Fisher information, from which the limit of precision can be obtained by Cramér-Rao bound. As a result, we find that Rayleigh's curse is present in such a scenario; however, SPADE measurement can outperform perfect direct imaging. What is more, a suitable adjustment of the measurement time between alignments allows measurement with near-optimal precision.
Due to the wave nature of light, optical imaging is inherently subject to diffraction, which typically dictates the resolution limit for two-point sources. A practical way to describe this phenomenon is through Rayleigh’s criterion, which states that two sources are barely resolvable when the minimum of one source's diffraction pattern coincides with the maximum of the other. However, recent advances in optical quantum metrology have shown that this classical limit can be surpassed by leveraging the spatial correlations within the light field. Specifically, decomposing the light into a Hermite-Gauss mode basis provides substantially more information than traditional direct imaging, which measures local intensity [1,2,3]. In this work, we implement a spatial demultiplexing technique to estimate the separation between light sources of unequal brightness. This unknown imbalance adds complexity to the estimation task, making it inherently a multiparameter problem and thus requiring a multimode approach [4]. By optimizing the measurement strategy and introducing a shift between two demultiplexers, we achieve sub-Rayleigh resolution in distance measurements without assuming prior knowledge of the intensity imbalance or alignment. [1] Tsang, M., Nair, R., & Lu, X. M. (2016). Quantum theory of superresolution for two incoherent optical point sources. Physical Review X, 6(3), 031033. [2] Sorelli, G., Gessner, M., Walschaers, M., & Treps, N. (2021). Optimal observables and estimators for practical superresolution imaging. Physical Review Letters, 127(12), 123604. [3] Rouvière, C., Barral, D., Grateau, A., Karuseichyk, I., Sorelli, G., Walschaers, M., & Treps, N. (2024). Ultra-sensitive separation estimation of optical sources. Optica, 11(2), 166-170. [4] Řeháček, J., Hradil, Z., Koutný, D., Grover, J., Krzic, A., & Sánchez-Soto, L. L. (2018). Optimal measurements for quantum spatial superresolution. Physical review A, 98(1), 012103.