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Maximum information states for coherent scattering measurements

An Author Correction to this article was published on 22 June 2021

This article has been updated


The use of coherent light for precision measurements has been a key driving force for numerous research directions, ranging from biomedical optics1,2 to semiconductor manufacturing3. Recent work demonstrates that the precision of such measurements can be substantially improved by tailoring the spatial profile of light fields used for estimating an observable system parameter4,5,6,7,8,9,10. These advances naturally raise the intriguing question of which states of light can provide the ultimate measurement precision11. Here we introduce a general approach to determine the optimal coherent states of light for estimating any given parameter, regardless of the complexity of the system. Our analysis reveals that the light fields delivering the ultimate measurement precision are eigenstates of a Hermitian operator that quantifies the Fisher information from the system’s scattering matrix12. To illustrate this concept, we experimentally show that these maximum information states can probe the phase or the position of an object that is hidden by a disordered medium with a precision improved by an order of magnitude compared with unoptimized states. Our results enable optimally precise measurements in arbitrarily complex systems, thus establishing a new benchmark for metrology and imaging applications3,13.

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Fig. 1: Principle of an optimal coherent scattering measurement.
Fig. 2: Characteristics of maximum information states.
Fig. 3: Demonstration of optimal estimations.

Data availability

Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

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  1. Park, Y., Depeursinge, C. & Popescu, G. Quantitative phase imaging in biomedicine. Nat. Photon. 12, 578–589 (2018).

    Article  ADS  Google Scholar 

  2. Taylor, R. W. & Sandoghdar, V. Interferometric scattering microscopy: seeing single nanoparticles and molecules via Rayleigh scattering. Nano Lett. 19, 4827–4835 (2019).

    Article  ADS  Google Scholar 

  3. Osten, W. & Reingand, N. Optical Imaging and Metrology: Advanced Technologies (John Wiley & Sons, 2012).

  4. van Putten, E. G., Lagendijk, A. & Mosk, A. P. Nonimaging speckle interferometry for high-speed nanometer-scale position detection. Opt. Lett. 37, 1070–1072 (2012).

    Article  ADS  Google Scholar 

  5. Shechtman, Y., Sahl, S. J., Backer, A. S. & Moerner, W. Optimal point spread function design for 3D imaging. Phys. Rev. Lett. 113, 133902 (2014).

    Article  ADS  Google Scholar 

  6. Balzarotti, F. et al. Nanometer resolution imaging and tracking of fluorescent molecules with minimal photon fluxes. Science 355, 606–612 (2017).

    Article  ADS  Google Scholar 

  7. Ambichl, P. et al. Super- and anti-principal-modes in multimode waveguides. Phys. Rev. X 7, 041053 (2017).

    Google Scholar 

  8. Yuan, G. H. & Zheludev, N. I. Detecting nanometric displacements with optical ruler metrology. Science 364, 771–775 (2019).

    Article  ADS  Google Scholar 

  9. Juffmann, T., de los Ríos Sommer, A. & Gigan, S. Local optimization of wave-fronts for optimal sensitivity phase imaging (LowPhi). Opt. Commun. 454, 124484 (2020).

    Article  Google Scholar 

  10. Bouchet, D., Carminati, R. & Mosk, A. P. Influence of the local scattering environment on the localization precision of single particles. Phys. Rev. Lett. 124, 133903 (2020).

    Article  ADS  Google Scholar 

  11. Giovannetti, V., Lloyd, S. & Maccone, L. Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011).

    Article  ADS  Google Scholar 

  12. Rotter, S. & Gigan, S. Light fields in complex media: mesoscopic scattering meets wave control. Rev. Mod. Phys. 89, 015005 (2017).

    Article  ADS  Google Scholar 

  13. Barrett, H. H. & Myers, K. J. Foundations of Image Science (John Wiley & Sons, 2013).

  14. Kay, S. Fundamentals of Statistical Processing Vol. I (Prentice Hall, 1993).

  15. Ambichl, P. et al. Focusing inside disordered media with the generalized Wigner-Smith operator. Phys. Rev. Lett. 119, 033903 (2017).

    Article  ADS  Google Scholar 

  16. Horodynski, M. et al. Optimal wave fields for micromanipulation in complex scattering environments. Nat. Photon. 14, 149–153 (2020).

    Article  ADS  Google Scholar 

  17. Mosk, A. P., Lagendijk, A., Lerosey, G. & Fink, M. Controlling waves in space and time for imaging and focusing in complex media. Nat. Photon. 6, 283–292 (2012).

    Article  ADS  Google Scholar 

  18. Fang, P., Zhao, L. & Tian, C. Concentration-of-measure theory for structures and fluctuations of waves. Phys. Rev. Lett. 121, 140603 (2018).

    Article  ADS  Google Scholar 

  19. Braunstein, S. L., Caves, C. M. & Milburn, G. J. Generalized uncertainty relations: theory, examples, and Lorentz invariance. Ann. Phys. 247, 135–173 (1996).

    Article  MathSciNet  ADS  Google Scholar 

  20. Zhou, E. H., Ruan, H., Yang, C. & Judkewitz, B. Focusing on moving targets through scattering samples. Optica 1, 227–232 (2014).

    Article  ADS  Google Scholar 

  21. Ma, C., Xu, X., Liu, Y. & Wang, L. V. Time-reversed adapted-perturbation (TRAP) optical focusing onto dynamic objects inside scattering media. Nat. Photon. 8, 931–936 (2014).

    Article  ADS  Google Scholar 

  22. Ruan, H. et al. Focusing light inside scattering media with magnetic-particle-guided wavefront shaping. Optica 4, 1337–1343 (2017).

    Article  ADS  Google Scholar 

  23. Carpenter, J., Eggleton, B. J. & Schröder, J. Observation of Eisenbud–Wigner–Smith states as principal modes in multimode fibre. Nat. Photon. 9, 751–757 (2015).

    Article  ADS  Google Scholar 

  24. Xiong, W. et al. Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling. Phys. Rev. Lett. 117, 053901 (2016).

    Article  ADS  Google Scholar 

  25. Hodaei, H. et al. Enhanced sensitivity at higher-order exceptional points. Nature 548, 187–191 (2017).

    Article  ADS  Google Scholar 

  26. Chen, W., Kaya Özdemir, S., Zhao, G., Wiersig, J. & Yang, L. Exceptional points enhance sensing in an optical microcavity. Nature 548, 192–196 (2017).

    Article  ADS  Google Scholar 

  27. Gérardin, B., Laurent, J., Derode, A., Prada, C. & Aubry, A. Full transmission and reflection of waves propagating through a maze of disorder. Phys. Rev. Lett. 113, 173901 (2014).

    Article  ADS  Google Scholar 

  28. Shi, Z. & Genack, A. Z. Transmission eigenvalues and the bare conductance in the crossover to Anderson localization. Phys. Rev. Lett. 108, 043901 (2012).

    Article  ADS  Google Scholar 

  29. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006).

    Article  MathSciNet  ADS  Google Scholar 

  30. Fiderer, L. J., Fraïsse, J. M. & Braun, D. Maximal quantum Fisher information for mixed states. Phys. Rev. Lett. 123, 250502 (2019).

    Article  MathSciNet  ADS  Google Scholar 

  31. Popoff, S. M. et al. Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media. Phys. Rev. Lett. 104, 100601 (2010).

    Article  ADS  Google Scholar 

  32. Pai, P., Bosch, J. & Mosk, A. P. Optical transmission matrix measurement sampled on a dense hexagonal lattice. OSA Contin. 3, 637–648 (2020).

    Article  Google Scholar 

  33. Takeda, M., Ina, H. & Kobayashi, S. Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry. J. Opt. Soc. Am. 72, 156–160 (1982).

    Article  ADS  Google Scholar 

  34. Cuche, E., Marquet, P. & Depeursinge, C. Spatial filtering for zero-order and twin-image elimination in digital off-axis holography. Appl. Opt. 39, 4070–4075 (2000).

    Article  ADS  Google Scholar 

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We thank M. van Beurden, J. Bosch, S. Faez, M. Horodynski, M. Kühmayer, P. Pai and J. Seifert for insightful discussions and P. Jurrius, D. Killian and C. de Kok for technical support. This work was supported by the Netherlands Organization for Scientific Research NWO (Vici grant number 68047618 and Perspective project number P16-08) and by the Austrian Science Fund (FWF) under project number P32300 (WAVELAND).

Author information

Authors and Affiliations



D.B. and A.P.M. initiated the project. D.B. performed the experiments. D.B., S.R. and A.P.M. developed the concept, analysed the results and wrote the manuscript.

Corresponding author

Correspondence to Dorian Bouchet.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Sébastien Popoff and Chushun Tian for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Optical setup.

The phases on the input SLM are modulated to reproduce the maximum information state, which reaches optimal sensitivity in its output with respect to any specified parameter characterizing the object displayed on the hidden SLM, such as phase variations or lateral displacements. Neutral density filters are removed for reflection matrix measurements. BS, beamsplitter; ND, neutral density filters; Obj, objective; NA, numerical aperture; Pol, linear polarizer; L1 and L2, lenses with focal length 200 mm.

Extended Data Fig. 2 Measured intensity and Fisher information distributions for maximum information states associated with different detection areas.

a-d, Measured spatial distributions of the intensity for optimal states, normalized by the average signal intensity under plane wave illumination. The observable parameter is the phase shift induced by the cross-shaped target, and optimal states are defined here with respect to a reduced field of view of the camera that covers an area of 220 μm2, as delimited by white dashed lines. e-h, Analogous to a-d for the measured spatial distribution of the Fisher information per unit area. Remarkably, the maximum information state always delivers the Fisher information to the designated observer window.

Source data

Extended Data Fig. 3 Single-pixel sensitivity for maximum information states associated with different observable parameters.

a-c, Single-pixel sensitivity measured by shifting the phase of each pixel in the target area of the hidden SLM for the optimal state, normalized by the average single-pixel sensitivity under plane-wave illumination. The object displayed on the hidden SLM is a circular phase object, whose position is delimited by a white dashed circle. The field of view of the detection camera covers here an area of 880 μm2. The incident states used to illuminate the scattering medium are the maximum information states relative to a phase shift (a), a horizontal shift (b) and a vertical shift (c) of the object. d-f, Analogous to a-c when the field of view of the camera covers a reduced area of 144 μm2. In all cases, the maximum information state directs the incoming intensity to those parts on the hidden SLM that are most affected by the change in the observable parameter. Interestingly, when the target parameter is either a horizontal or a vertical shift of the object, the maximum information state typically focuses on a single edge rather than on both edges simultaneously, which is to be expected as the mirror symmetry in the system is broken by the diffuser.

Source data

Extended Data Fig. 4 Predicted and measured signal intensity distribution at low photon counts.

a, Predicted distribution of the signal intensity for the optimal incident state expressed in analog-to-digital units (ADU). This distribution is calculated from the measured reflection matrix, considering that the neutral density filter ND6 (fractional transmittance 8.3 × 10−7) is placed in the optical path. b, Measured distribution of the signal intensity for the optimal incident state when ND6 is in the signal path. Such measurements are shot-noise limited, and the observed signal-to-noise per pixel is largely smaller than unity. Thus, the measured distribution of the signal intensity appears as a random noise, which has been low-pass filtered by the data analysis procedure used to digitally reconstruct complex fields from off-axis intensity measurements. Despite this low signal-to-noise per pixel, such data allow to correctly estimate the phase shift induced by the hidden target when using the minimum variance unbiased estimator. This can be achieved since only a single parameter (the phase shift induced by the target) needs to be estimated from a large number of independent sampling points. The Fisher information associated with each pixel of the detection camera effectively adds up, resulting in a total Fisher information that is sufficient to resolve the phase steps induced on the hidden SLM. c, d, Analogous to a and b for the best plane wave used to construct the reflection matrix.

Source data

Extended Data Fig. 5 Estimations of lateral displacements at low photon counts.

Analogous to Fig. 3 when the observable parameter is the horizontal position x of the circular phase object shown in Fig. 2i. a, Histogram of precision limits for the 2,437 plane waves used to construct the reflection matrix and for the maximum information states (AP, amplitude and phase modulation; PO, phase-only modulation). b, Estimated lateral displacements of the circular phase object as a function of measurement index for measurements performed by illuminating the medium with the maximum information state. The calculated precision limit equals 1.4 μm, and the observed standard error on the estimates is 1.5 μm. c, Histograms of estimated angles for a positive lateral displacement (Δx+ = + 2.5 μm) and a negative lateral displacement (Δx = − 2.5 μm) applied by the hidden SLM. The length of error bars equals 2 σCRB. d, e, Analogous to b and c for measurements performed by illuminating the medium with the best plane wave.

Source data

Supplementary information

Supplementary Information

Supplementary Sections 1–3.

Source data

Source Data Fig. 2

Intensity and Fisher information distributions for maximum information states.

Source Data Fig. 3

Precision limit for plane waves and estimations of phase shift.

Source Data Extended Data Fig. 2

Intensity and Fisher information distributions for different detection areas.

Source Data Extended Data Fig. 3

Single-pixel sensitivity for different observable parameters.

Source Data Extended Data Fig. 4

Predicted and measured intensity distribution at low photon counts.

Source Data Extended Data Fig. 5

Precision limit for plane waves and estimations of lateral displacement.

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Bouchet, D., Rotter, S. & Mosk, A.P. Maximum information states for coherent scattering measurements. Nat. Phys. 17, 564–568 (2021).

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