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Chinese Optics Letters

Quantum-limited resolution of partially coherent sources

Ben Wang, Liang Xu, Hongkuan Xia, Aonan Zhang, Kaimin Zheng, Lijian Zhang

Published Jan 1, 2023• 6 citations
Publisher

Abstract

Discriminating two spatially separated sources is one of the most fundamental problems in imaging.Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken.However, in realistic optical systems, there often exists coherence in the imaging light field, and there have been efforts to analyze the optical resolution in the presence of partial coherence.Nevertheless, how the degree of coherence between two point sources affects the resolution has not been fully understood.Here, we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources.In particular, we consider the situation in which coherence varies with the separation.We propose a feasible experiment scheme to realize the nearly optimal measurement, which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging.Our results will have wide applications in imaging involving coherence of light.

DOI
10.3788/col202321.042601

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Discriminating two spatially separated sources is one of the most fundamental problems in imaging.Recent research based on quantum parameter estimation theory shows that the resolution limit of two incoherent point sources given by Rayleigh can be broken.However, in realistic optical systems, there often exists coherence in the imaging light field, and there have been efforts to analyze the optical resolution in the presence of partial coherence.Nevertheless, how the degree of coherence between two point sources affects the resolution has not been fully understood.Here, we analyze the quantum-limited resolution of two partially coherent point sources by explicitly relating the state after evolution through the optical systems to the coherence of the sources.In particular, we consider the situation in which coherence varies with the separation.We propose a feasible experiment scheme to realize the nearly optimal measurement, which adaptively chooses the binary spatial-mode demultiplexing measurement and direct imaging.Our results will have wide applications in imaging involving coherence of light.
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].
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 (see Fig. 1 (left)) 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].
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. However, most previous works have either consider low-intensity, or thermal sources. To broaden the applicability of this approach, it is important to extend these results to more general light sources. To this goal, we will present an analytical expression for the Quantum Fisher Information, determining the ultimate resolution limit, for the separation between two sources in an arbitrary Gaussian state. Applying this result to different quantum states, we can shine some light on some relevant questions. We can for example explore the role of partial coherence considering displaced and correlated thermal states, or investigate the importance of quantum correlations by considering squeezed light. In addition to the ultimate quantum limit, we will discuss a simple estimation technique, requiring access only to the mean value of a linear combination of demultiplexed intensity measurements, which is often sufficient to saturate these limits, and can easily be adapted to incorporate the most common noise sources. Finally, we will present our experimental setup that allows for the generation of the images of two sources with different photon statistics, as well as for spatial mode demultiplexing and we will discuss the first practical implementations if the above mentioned estimation techniques.
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.
The two-point resolution of an optical system is the minimum distance between two point sources that can be estimated with a prescribed precision from measurement in the image plane. When the sources are incoherent, then direct measurement of the optical intensity provides resolution limited by Rayleigh’s curse, i.e., the precision diminishes to zero as the separation is reduced to zero. By using quantum Fisher information bounds on the precision, it was shown recently that estimates based on optimal quantum measurements of the optical field can break Rayleigh’s curse and provide estimates with finite precision even at very small separations. We show here that if the point sources are partially coherent with an unknown real degree of coherence, no matter how small it is, then the curse resurges. Since a Lambertian source is not strictly incoherent, having a correlation width of the order of a wavelength, and light gains coherence as it propagates, Rayleigh’s curse endures as a fundamental dictum.
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.