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Gutiérrez-Cuevas R, Bouchet D, de Rosny J, Popoff SM. Reaching the precision limit with tensor-based wavefront shaping. Nat Commun 2024; 15:6319. [PMID: 39060250 PMCID: PMC11282273 DOI: 10.1038/s41467-024-50513-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2023] [Accepted: 07/14/2024] [Indexed: 07/28/2024] Open
Abstract
Perturbations in complex media, due to their own dynamical evolution or to external effects, are often seen as detrimental. Therefore, a common strategy, especially for telecommunication and imaging applications, is to limit the sensitivity to those perturbations in order to avoid them. Here, instead, we consider enhancing the interaction between light and perturbations to produce the largest change in the output intensity distribution. Our work hinges on the use of tensor-based techniques, presently at the forefront of machine learning explorations, to study intensity-based measurements where its quadratic relationship to the field prevents the use of standard matrix methods. With this tensor-based framework, we can identify the maximum-information intensity channel which maximizes the change in its output intensity distribution and the Fisher information encoded in it about a given perturbation. We further demonstrate experimentally its superiority for robust and precise sensing applications. Additionally, we derive the appropriate strategy to reach the precision limit for intensity-based measurements, leading to an increase in Fisher information by more than four orders of magnitude compared to the mean for random wavefronts when measured with the pixels of a camera.
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Affiliation(s)
| | - Dorian Bouchet
- Univ. Grenoble Alpes, CNRS, LIPhy, 38000, Grenoble, France
| | - Julien de Rosny
- Institut Langevin, ESPCI Paris, Université PSL, CNRS, 75005, Paris, France
| | - Sébastien M Popoff
- Institut Langevin, ESPCI Paris, Université PSL, CNRS, 75005, Paris, France
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2
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Bouchet D, Rachbauer LM, Rotter S, Mosk AP, Bossy E. Optimal Control of Coherent Light Scattering for Binary Decision Problems. PHYSICAL REVIEW LETTERS 2021; 127:253902. [PMID: 35029434 DOI: 10.1103/physrevlett.127.253902] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2021] [Accepted: 11/18/2021] [Indexed: 05/25/2023]
Abstract
Because of quantum noise fluctuations, the rate of error achievable in decision problems involving several possible configurations of a scattering system is subject to a fundamental limit known as the Helstrom bound. Here, we present a general framework to calculate and minimize this bound using coherent probe fields with tailored spatial distributions. As an example, we experimentally study a target located in between two disordered scattering media. We first show that the optimal field distribution can be directly identified using a general approach based on scattering matrix measurements. We then demonstrate that this optimal light field successfully probes the presence of the target with a number of photons that is reduced by more than 2 orders of magnitude as compared to unoptimized fields.
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Affiliation(s)
- Dorian Bouchet
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
| | - Lukas M Rachbauer
- Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
| | - Stefan Rotter
- Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
| | - Allard P Mosk
- Nanophotonics, Debye Institute for Nanomaterials Science and Center for Extreme Matter and Emergent Phenomena, Utrecht University, P.O. Box 80000, 3508 TA Utrecht, Netherlands
| | - Emmanuel Bossy
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
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3
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Horodynski M, Bouchet D, Kühmayer M, Rotter S. Invariance Property of the Fisher Information in Scattering Media. PHYSICAL REVIEW LETTERS 2021; 127:233201. [PMID: 34936787 DOI: 10.1103/physrevlett.127.233201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Accepted: 10/27/2021] [Indexed: 06/14/2023]
Abstract
Determining the ultimate precision limit for measurements on a subwavelength particle with coherent laser light is a goal with applications in areas as diverse as biophysics and nanotechnology. Here, we demonstrate that surrounding such a particle with a complex scattering environment does, on average, not have any influence on the mean quantum Fisher information associated with measurements on the particle. As a remarkable consequence, the average precision that can be achieved when estimating the particle's properties is the same in the ballistic and in the diffusive scattering regime, independently of the particle's position within its nonabsorbing environment. This invariance law breaks down only in the regime of Anderson localization, due to increased C_{0}-speckle correlations. Finally, we show how these results connect to the mean quantum Fisher information achievable with spatially optimized input fields.
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Affiliation(s)
- Michael Horodynski
- Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
| | - Dorian Bouchet
- Université Grenoble Alpes, CNRS, LIPhy, 38000 Grenoble, France
| | - Matthias Kühmayer
- Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
| | - Stefan Rotter
- Institute for Theoretical Physics, Vienna University of Technology (TU Wien), 1040 Vienna, Austria
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4
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Zheltikov AM. Imaging through a scattering medium: the Fisher information and the generalized Abbe limit. OPTICS LETTERS 2021; 46:5902-5905. [PMID: 34851919 DOI: 10.1364/ol.439132] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2021] [Accepted: 10/07/2021] [Indexed: 06/13/2023]
Abstract
Enhanced-resolution imaging in complex scattering media is revisited from a parameter estimation perspective. A suitably defined Fisher information is shown to offer useful insights into the limiting precision of parameter estimation in a scattering environment and, hence, into the limiting spatial resolution that can be achieved in imaging-through-scattering settings. The Fisher information that defines this resolution limit via the Cramér-Rao lower bound is shown to scale with the number of adaptively controlled space-time modes of the probe field, suggesting a physically intuitive generalization of the Abbe limit to the spatial resolution attainable for complex scattering systems. In a conventional, direct-imaging microscopy setting, this bound is shown to converge to the canonical Abbe limit.
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Del Hougne M, Gigan S, Del Hougne P. Deeply Subwavelength Localization with Reverberation-Coded Aperture. PHYSICAL REVIEW LETTERS 2021; 127:043903. [PMID: 34355940 DOI: 10.1103/physrevlett.127.043903] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/10/2021] [Revised: 06/10/2021] [Accepted: 06/14/2021] [Indexed: 05/18/2023]
Abstract
Accessing subwavelength information about a scene from the far-field without invasive near-field manipulations is a fundamental challenge in wave engineering. Yet it is well understood that the dwell time of waves in complex media sets the scale for the waves' sensitivity to perturbations. Modern coded-aperture imagers leverage the degrees of freedom (d.o.f.) offered by complex media as natural multiplexor but do not recognize and reap the fundamental difference between placing the object of interest outside or within the complex medium. Here, we show that the precision of localizing a subwavelength object can be improved by several orders of magnitude simply by enclosing it in its far field with a reverberant passive chaotic cavity. We identify deep learning as a suitable noise-robust tool to extract subwavelength localization information encoded in multiplexed measurements, achieving resolutions well beyond those available in the training data. We demonstrate our finding in the microwave domain: harnessing the configurational d.o.f. of a simple programmable metasurface, we localize a subwavelength object along a curved trajectory inside a chaotic cavity with a resolution of λ/76 using intensity-only single-frequency single-pixel measurements. Our results may have important applications in photoacoustic imaging as well as human-machine interaction based on reverberating elastic waves, sound, or microwaves.
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Affiliation(s)
| | - Sylvain Gigan
- Laboratoire Kastler Brossel, Université Pierre et Marie Curie, Ecole Normale Supérieure, CNRS, Collège de France, F-75005 Paris, France
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Wei X, Urbach HP, van der Walle P, Coene WMJ. Parameter retrieval of small particles in dark-field Fourier ptychography and a rectangle in real-space ptychography. Ultramicroscopy 2021; 229:113335. [PMID: 34243020 DOI: 10.1016/j.ultramic.2021.113335] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2020] [Revised: 03/08/2021] [Accepted: 05/18/2021] [Indexed: 10/21/2022]
Abstract
We present a parameter retrieval method which incorporates prior knowledge about the object into ptychography. The proposed method is applied to two applications: (1) parameter retrieval of small particles from Fourier ptychographic dark field measurements; (2) parameter retrieval of a rectangular structure with real-space ptychography. The influence of Poisson noise is discussed in the second part of the paper. The Cramér Rao Lower Bound in both applications is computed and Monte Carlo analysis is used to verify the calculated lower bound. With the computation results we report the lower bound for various noise levels and analyze the correlation of particles in application 1. For application 2 the correlation of parameters of the rectangular structure is discussed.
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Affiliation(s)
- Xukang Wei
- Optics Research Group, Delft University of Technology, Lorentzweg 1, Delft, 2628 CJ, The Netherlands.
| | - H Paul Urbach
- Optics Research Group, Delft University of Technology, Lorentzweg 1, Delft, 2628 CJ, The Netherlands
| | | | - Wim M J Coene
- Optics Research Group, Delft University of Technology, Lorentzweg 1, Delft, 2628 CJ, The Netherlands; ASML Netherlands B.V, De Run 6501, Veldhoven, 5504 DR, The Netherlands
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Bouchet D, Seifert J, Mosk AP. Optimizing illumination for precise multi-parameter estimations in coherent diffractive imaging. OPTICS LETTERS 2021; 46:254-257. [PMID: 33449001 DOI: 10.1364/ol.411339] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 11/30/2020] [Indexed: 06/12/2023]
Abstract
Coherent diffractive imaging (CDI) is widely used to characterize structured samples from measurements of diffracting intensity patterns. We introduce a numerical framework to quantify the precision that can be achieved when estimating any given set of parameters characterizing the sample from measured data. The approach, based on the calculation of the Fisher information matrix, provides a clear benchmark to assess the performance of CDI methods. Moreover, by optimizing the Fisher information metric using deep learning optimization libraries, we demonstrate how to identify the optimal illumination scheme that minimizes the estimation error under specified experimental constraints. This work paves the way for an efficient characterization of structured samples at the sub-wavelength scale.
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Pai P, Bosch J, Mosk AP. Resampling the transmission matrix in an aberration-corrected Bessel mode basis. OPTICS EXPRESS 2021; 29:24-36. [PMID: 33362098 DOI: 10.1364/oe.412540] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/14/2020] [Accepted: 12/06/2020] [Indexed: 06/12/2023]
Abstract
The study of the optical transmission matrix (TM) of a sample reveals important statistics of light transport through it. The accuracy of the statistics depends strongly on the orthogonality and completeness of the basis in which the TM is measured. While conventional experimental methods suffer from sampling effects and optical aberrations, we use a basis of Bessel modes of the first kind to faithfully recover the singular values, eigenvalues and eigenmodes of light propagation through a finite thickness of air.
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Buijs R, Schilder NJ, Wolterink TAW, Gerini G, Verhagen E, Koenderink AF. Super-Resolution without Imaging: Library-Based Approaches Using Near-to-Far-Field Transduction by a Nanophotonic Structure. ACS PHOTONICS 2020; 7:3246-3256. [PMID: 33241077 PMCID: PMC7678721 DOI: 10.1021/acsphotonics.0c01350] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/28/2020] [Indexed: 06/11/2023]
Abstract
Super-resolution imaging is often viewed in terms of engineering narrow point spread functions, but nanoscale optical metrology can be performed without real-space imaging altogether. In this paper, we investigate how partial knowledge of scattering nanostructures enables extraction of nanoscale spatial information from far-field radiation patterns. We use principal component analysis to find patterns in calibration data and use these patterns to retrieve the position of a point source of light. In an experimental realization using angle-resolved cathodoluminescence, we retrieve the light source position with an average error below λ/100. The patterns found by principal component analysis reflect the underlying scattering physics and reveal the role the scattering nanostructure plays in localization success. The technique described here is highly general and can be applied to gain insight into and perform subdiffractive parameter retrieval in various applications.
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Affiliation(s)
- Robin
D. Buijs
- Center
for Nanophotonics, AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands
| | - Nick J. Schilder
- Center
for Nanophotonics, AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands
| | - Tom A. W. Wolterink
- Center
for Nanophotonics, AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands
| | - Giampiero Gerini
- Optics
Department, Netherlands Organization for
Applied Scientific Research (TNO), Stieltjesweg 1, 2628CK Delft, The Netherlands
- Department
of Electrical Engineering, Technische Universiteit
Eindhoven (TU/e), 5600MB Eindhoven, The Netherlands
| | - Ewold Verhagen
- Center
for Nanophotonics, AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands
| | - A. Femius Koenderink
- Center
for Nanophotonics, AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands
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