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Physica Scripta

Achieving spatial superresolution with engineered spatial modes

Jerin A. Thachil, B.S. Ramanan, Ashok Kumar

Published Oct 13, 2023• 2 citations

Abstract

Abstract Rayleigh’s criterion sets a limit on the minimum separation between two incoherent point sources to be resolved into distinct objects. However, superresolution techniques have been developed to circumvent Rayleigh’s criterion. These techniques mainly deal with single parameter estimation and require prior information about the centroid. Here, we use multi-parameter estimation tools to simultaneously and optimally retrieve information about the centroid and object separation. Collective measurements on photons using two-photon interference followed by spatially resolved detection have significantly improved over direct detection schemes. Following the same approach, we extend the analysis of the two-photon interference protocol to spatially engineered photons having a Pearson type VII profile with arbitrary positive excess kurtosis. We calculate the precision limits in the current measurement scheme as well as the ultimate precision limits based on the quantum Cramer–Rao bound for different spatial modes. We theoretically show that such engineered pulses show enhanced precision with increasing kurtosis in simultaneous estimation of the centroid and object separation compared to a Gaussian amplitude profile. Furthermore, we discuss an experimental setup to realize the proposed superresolution scheme.

DOI: 10.1088/1402-4896/ad0344

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Multiparameter estimation theory offers a general framework to explore imaging techniques beyond the Rayleigh limit. While optimal measurements of single parameters characterizing a composite light source are now well understood, simultaneous determination of multiple parameters poses a much greater challenge that in general requires implementation of collective measurements. Here we show, theoretically and experimentally, that Hong-Ou-Mandel interference followed by spatially resolved detection of photons provides precise information on both the separation and the centroid for a pair of point emitters, avoiding trade-offs inherent to single-photon measurements.

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.

We establish the multiparameter quantum Cram\'er-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cram\'er-Rao bound is independent of their separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost importance in realistic scenarios, such as observational astronomy.

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.

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.

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