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arXiv:2312.11044 · Quantum Frontiers

Experimental 3D super-localization with Laguerre–Gaussian modes

Chenyu Hu, Liang Xu, Ben Wang, Zhiwen Li, Yipeng Zhang, Yong Zhang, Lijian Zhang

Published Nov 29, 2023• 5 citations
arXivHTMLPDF

Abstract

Abstract Improving three-dimensional (3D) localization precision is of paramount importance for super-resolution imaging. By properly engineering the point spread function (PSF), such as utilizing Laguerre–Gaussian (LG) modes and their superposition, the ultimate limits of 3D localization precision can be enhanced. However, achieving these limits is challenging, as it often involves complicated detection strategies and practical limitations. In this work, we rigorously derive the ultimate 3D localization limits of LG modes and their superposition, specifically rotation modes, in the multi-parameter estimation framework. Our findings reveal that a significant portion of the information required for achieving 3D super-localization of LG modes can be obtained through feasible intensity detection. Moreover, the 3D ultimate precision can be achieved when the azimuthal index l is zero. To provide a proof-of-principle demonstration, we develop an iterative maximum likelihood estimation (MLE) algorithm that converges to the 3D position of a point source, considering the pixelation and detector noise. The experimental implementation exhibits an improvement of up to two-fold in lateral localization precision and up to twenty-fold in axial localization precision when using LG modes compared to Gaussian mode. We also showcase the superior axial localization capability of the rotation mode within the near-focus region, effectively overcoming the limitations encountered by single LG modes. Notably, in the presence of realistic aberration, the algorithm robustly achieves the Cramér-Rao lower bound. Our findings provide valuable insights for evaluating and optimizing the achievable 3D localization precision, which will facilitate the advancements in super-resolution microscopy.

DOI
10.1007/s44214-023-00047-9

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Abstract Improving three-dimensional (3D) localization precision is of paramount importance for super-resolution imaging. By properly engineering the point spread function (PSF), such as utilizing Laguerre–Gaussian (LG) modes and their superposition, the ultimate limits of 3D localization precision can be enhanced. However, achieving these limits is challenging, as it often involves complicated detection strategies and practical limitations. In this work, we rigorously derive the ultimate 3D localization limits of LG modes and their superposition, specifically rotation modes, in the multi-parameter estimation framework. Our findings reveal that a significant portion of the information required for achieving 3D super-localization of LG modes can be obtained through feasible intensity detection. Moreover, the 3D ultimate precision can be achieved when the azimuthal index l is zero. To provide a proof-of-principle demonstration, we develop an iterative maximum likelihood estimation (MLE) algorithm that converges to the 3D position of a point source, considering the pixelation and detector noise. The experimental implementation exhibits an improvement of up to two-fold in lateral localization precision and up to twenty-fold in axial localization precision when using LG modes compared to Gaussian mode. We also showcase the superior axial localization capability of the rotation mode within the near-focus region, effectively overcoming the limitations encountered by single LG modes. Notably, in the presence of realistic aberration, the algorithm robustly achieves the Cramér-Rao lower bound. Our findings provide valuable insights for evaluating and optimizing the achievable 3D localization precision, which will facilitate the advancements in super-resolution microscopy.
Further improving the axial resolution is paramount for three-dimensional optical imaging systems. Vortex beams are being widely applied in 3D microscopy techniques. Here we theoretically investigate the ultimate resolution limits using Laguerre-Gauss (LG) beams. Various kinds of superpositions can nowadays be easily prepared by spatial light modulators (SLM). It has been keenly shown that LG beams’ superpositions possess more information than pure LG beams yet do not saturate the ultimate limit with a simple intensity scan. More sophisticated detection schemes based on quantum super-resolution protocols are investigated here to retrieve the discarded information.
We investigate the localization of two incoherent point sources with arbitrary angular and axial separations in the paraxial approximation. By using quantum metrology techniques, we show that a simultaneous estimation of the two separations is achievable by a single quantum measurement, with a precision saturating the ultimate limit stemming from the quantum Cramér-Rao bound. Such a precision is not degraded in the subwavelength regime, thus overcoming the traditional limitations of classical direct imaging derived from Rayleigh's criterion. Our results are qualitatively independent of the point spread function of the imaging system, and quantitatively illustrated in detail for the Gaussian instance. This analysis may have relevant applications in three-dimensional surface measurements.
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In this work we experimentally demonstrate high sensitive strictly-local identification of azimuthal index of Laguerre–Gauss (LG) beams, with less than 160 photon counts. To this aim, detection of the azimuthal index of LG beams is performed with an innovative interferometer relying on a monolithic birefringent crystal, thus ensuring stability without the need of any feedback or thermal drift compensation. By first generating a reference interference pattern with a standard TEM00 mode, we then detect the value of the azimuthal index of a LG beam from the lateral shift of the pattern with respect to the reference one. An experimental setup has been realized to prove the effectiveness of the proposed scheme, which requires to access only a small portion (5%) of the entire wavefront. Moreover, being intrinsically endowed with extreme robustness and stability, we achieve effective high sensitive detection of the azimuthal index by collecting less than 160 photons only, while at the same time keeping the local features. Limitations and possible applications are also discussed.
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