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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Temporal Resolution of Partially Coherent Sources

Syamsundar De, Jano Gil-López, Benjamin Brecht, Christine Silberhorn, Luis Lorenzo Sánchez-Soto, Z. Hradil, Jaroslav Řeháček

Published Jun 21, 2021• 0 citations
Publisher

Abstract

In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.

DOI
10.1109/cleo/europe-eqec52157.2021.9541755

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In spatial imaging, it is well known that two incoherent sources with a separation less than the diffraction limit cannot be resolved, a phenomenon dubbed as Rayleigh’s curse. A recent information theoretic approach suggests that this curse can be avoided using optimal (here, mode-sensitive) measurements instead of the usual intensity-only measurements, which define the classical limit [1] . This quantum enhanced resolution has been realized in both the spatial [2] and temporal domain [3] . Nevertheless, the role of optical coherence on the resolution limit still remains elusive [4] , [5] , [6] . Here, we experimentally resolve this debate using time-domain measurements.
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.
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.
Every imaging system has a resolution limit, typically defined by Rayleigh's criterion. Given a fixed number of photons, the amount of information one can gain from an image about the separation between two sources falls to zero as the separation drops below this limit, an effect dubbed "Rayleigh's curse." Recently, in a quantum-information-inspired proposal, Tsang and co-workers found that there is, in principle, infinitely more information present in the full electromagnetic field in the image plane than in the intensity alone, and suggested methods for extracting this information and beating the Rayleigh limit. In this Letter, we experimentally demonstrate a simple scheme that captures most of this information, and show that it has a greatly improved ability to estimate the distance between a pair of closely separated sources, achieving near-quantum-limited performance and immunity to Rayleigh's curse.
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.
The resolution limit of standard imaging techniques is expressed by the Rayleigh criterion, which states that two point-like sources are difficult to resolve if their transverse separation is smaller that the Rayleigh length. While the criterion is useful in the case of direct detection imaging, other measurement techniques may not be subject to this limitation. Here we consider the use of imaging to estimate the distance between two incoherent point sources. In this framework, the Rayleigh criterion manifests itself through the so-called Rayleigh curse. The latter expresses the fact that the statistical error in the estimation increases for sub-wavelength separation. Here we show that, in the regime of weak signals, a structured measurement obtained by concatenating a linear interferometer with on-off photo-detection is immune to the Rayleigh curse. In this way we clarify the relation between imaging and interferometry, and establish the optimality of linear interferometry for an arbitrary number of incoherent sources.
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.