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arXiv:2608.07868

Sub-Rayleigh Imaging of Unequal-Intensity Sources: Near-Quantum-Limit Multiparameter Estimation

Y. Batuhan Yilmaz, Kent A. G. Bonsma-Fisher, Muhammad Mohid, Noah Lupu-Gladstein, Arthur O. T. Pang, Aephraim M. Steinberg

physics.opticsquant-phSubmitted Aug 8, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

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.

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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.
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.
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.
Superresolution techniques based on intensity measurements after a spatial mode decomposition can overcome the precision of diffraction-limited direct imaging. We present both the experimental implementation of simultaneous spatial multimode demultiplexing as a distance measurement tool and the theoretical analysis of the actual sensitivity limits given the main experimental imperfection: cross-talks between channels. We demonstrate the distance estimation between two incoherent beams in both directions of the transverse plane, and find a perfect accordance with theoretical predictions, given a proper calibration of the demultiplexer. We show that, even though sensitivity is limited by the cross-talk between channels, we can perform measurements in two dimensions much beyond the Rayleigh limit over a large dynamic range. Combining statistical and analytical tools, we obtain the scaling of the precision limits for weak, generic crosstalk from a device-independent model as a function of the crosstalk probability and N.
Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrology-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite–Gauss basis. Analyzing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.
Historically, the resolution of optical imaging systems was dictated by diffraction, and the Rayleigh criterion was long considered an unsurpassable limit. In superresolution microscopy, this limit is overcome by manipulating the emission properties of the object. However, in passive imaging, when sources are uncontrolled, reaching sub-Rayleigh resolution remains a challenge. Here, we implement a quantum-metrolgy-inspired approach for estimating the separation between two incoherent sources, achieving a sensitivity five orders of magnitude beyond the Rayleigh limit. Using a spatial mode demultiplexer, we examine scenes with bright and faint sources, through intensity measurements in the Hermite-Gauss basis. Analysing sensitivity and accuracy over an extensive range of separations, we demonstrate the remarkable effectiveness of demultiplexing for sub-Rayleigh separation estimation. These results effectively render the Rayleigh limit obsolete for passive imaging.
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.