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.
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.
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.
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.
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.
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.
In the context of separation estimation between two incoherent point sources, it has recently been shown that an optimal measurement strategy, which saturates the quantum Cramer-Rao bound, involves the use of spatial mode demultiplexing method (SPADE). To realize mode selective measurement required for SPADE, we propose a new approach based on sum frequency generation (SFG). The conversion of infrared light coming from two incoherent point sources is performed in a periodically-poled lithium niobate (PPLN) crystal by mean of a spatially shaped pump laser. By analyzing the converted images obtained with pump beams shaped as Hermite-Gaussian (HG) modes, we demonstrate the mode-sorting capabilities of this system. Our experiment, shows that our measurement method can estimate separations in sub-Rayleigh regime with improved accuracy compared to the traditional direct imaging method.
Superresolution is one of the key issues at the crossroads of contemporary quantum optics and metrology. Recently, it was shown that for an idealized case of two balanced sources, spatial mode demultiplexing (SPADE) achieves resolution better than direct imaging even in the presence of measurement crosstalk [Phys. Rev. Lett. 125, 100501 (2020)]. In this work, we consider arbitrarily unbalanced sources and provide a systematic analysis of the impact of crosstalk on the resolution obtained from SPADE. As we dissect, in this generalized scenario, SPADE's effectiveness depends non-trivially on the strength of crosstalk, relative brightness and the separation between the sources. In particular, for any source imbalance, SPADE performs worse than ideal direct imaging in the asymptotic limit of vanishing source separations. Nonetheless, for realistic values of crosstalk strength, SPADE is still the superior method for several orders of magnitude of source separations.