Progress in imaging and metrology depends on exquisite control over and comprehensive characterization of wave fields. As reflected in its name, coherent diffractive imaging relies on high coherence when reconstructing highly resolved images from diffraction intensities alone without the need for image-forming lenses1,2,3. Fully coherent light can be described adequately by a single pure state. Yet partial coherence and imperfect detection often need to be accounted for, requiring statistical optics or the superposition of states4,5. Furthermore, the dynamics of samples are increasingly the very objectives of experiments6. Here we provide a general analytic approach to the characterization of diffractive imaging systems that can be described as low-rank mixed states. We use experimental data and simulations to show how the reconstruction technique compensates for and characterizes various sources of decoherence quantitatively. Based on ptychography7,8, the procedure is closely related to quantum state tomography and is equally applicable to high-resolution microscopy, wave sensing and fluctuation measurements. As a result, some of the most stringent experimental conditions in ptychography can be relaxed, and susceptibility to imaging artefacts is reduced. Furthermore, the method yields high-resolution images of mixed states within the sample, which may include quantum mixtures or fast stationary stochastic processes such as vibrations, switching or steady flows.
This is a preview of subscription content, access via your institution
Open Access articles citing this article.
PhotoniX Open Access 24 January 2023
Scientific Reports Open Access 24 August 2022
Scientific Reports Open Access 19 July 2022
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Get time limited or full article access on ReadCube.
All prices are NET prices.
Miao, J., Charalambous, P., Kirz, J. & Sayre, D. Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature 400, 342–344 (1999)
Eisebitt, S. et al. Lensless imaging of magnetic nanostructures by X-ray spectro-holography. Nature 432, 885–888 (2004)
Abbey, B. et al. Keyhole coherent diffractive imaging. Nature Phys. 4, 394–398 (2008)
Mandel, L. & Wolf, E. Coherence properties of optical fields. Rev. Mod. Phys. 37, 231–287 (1965)
Goodman, J. W. Statistical Optics (Wiley, 2000)
Sutton, M. A review of X-ray intensity fluctuation spectroscopy. C. R. Phys. 9, 657–667 (2008)
Thibault, P. et al. High-resolution scanning X-ray diffraction microscopy. Science 321, 379–382 (2008)
Maiden, A. M. & Rodenburg, J. M. An improved ptychographical phase retrieval algorithm for diffractive imaging. Ultramicroscopy 109, 1256–1262 (2009)
Clark, J. N. & Peele, A. G. Simultaneous sample and spatial coherence characterisation using diffractive imaging. Appl. Phys. Lett. 99, 154103 (2011)
Dierolf, M. et al. Ptychographic X-ray computed tomography at the nanoscale. Nature 467, 436–439 (2010)
Putkunz, C. et al. Atom-scale ptychographic electron diffractive imaging of boron nitride cones. Phys. Rev. Lett. 108, 1–4 (2012)
Maiden, A. M., Rodenburg, J. M. & Humphry, M. J. Optical ptychography: a practical implementation with useful resolution. Opt. Lett. 35, 2585–2587 (2010)
von Neumann, J. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 245–272 (1927)
Wolf, E. New theory of partial coherence in the space-frequency domain Part I: spectra and cross spectra of steady-state sources. J. Opt. Soc. Am. 72, 343–351 (1982)
Whitehead, L. W. et al. Diffractive imaging using partially coherent X rays. Phys. Rev. Lett. 103, 243902 (2009)
Abbey, B. et al. Lensless imaging using broadband X-ray sources. Nature Photon. 5, 420–424 (2011)
Candès, E. J. & Recht, B. Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)
Gross, D. Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57, 1548–1566 (2011)
Rodenburg, J. M. & Bates, R. H. T. The theory of super-resolution electron microscopy via Wigner-distribution deconvolution. Phil. Trans. R. Soc. Lond. A 339, 521–553 (1992)
Chapman, H. N. Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution: signal processing. Scanning Microsc. 11, 67–80 (1997)
Raymer, M. G. Measuring the quantum mechanical wave function. Contemp. Phys. 38, 343–355 (1997)
Fienup, J. R. Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758–2769 (1982)
Elser, V. Phase retrieval by iterated projections. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 20, 40–55 (2003)
Guizar-Sicairos, M. & Fienup, J. R. Phase retrieval with transverse translation diversity: a nonlinear optimization approach. Opt. Express 16, 7264–7278 (2008)
Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. N. J. Phys. 14, 063004 (2012)
Pfeiffer, F. et al. Shearing interferometer for quantifying the coherence of hard X-ray beams. Phys. Rev. Lett. 94, 164801 (2005)
Cerbino, R. et al. X-ray-scattering information obtained from near-field speckle. Nature Phys. 4, 238–243 (2008)
Clark, J. N. et al. Dynamic sample imaging in coherent diffractive imaging. Opt. Lett. 36, 1954–1956 (2011)
Cowley, J. M. Image contrast in a transmission scanning electron microscope. Appl. Phys. Lett. 58, 58–59 (1969)
Treacy, M. M. J., Gibson, J. M., Fan, L., Paterson, D. J. & McNulty, I. Fluctuation microscopy: a probe of medium range order. Rep. Prog. Phys. 68, 2899–2944 (2005)
Gorelick, S., Guzenko, V. A., Vila-Comamala, J. & David, C. Direct e-beam writing of dense and high aspect ratio nanostructures in thick layers of PMMA for electroplating. Nanotechnology 21, 295303 (2010)
Henrich, B. et al. PILATUS: a single photon counting pixel detector for x-ray applications. Nucl. Instrum. Methods Phys. Res. A 607, 247–249 (2009)
We thank M. Dierolf for discussions and help in the algorithm implementation; A. Diaz for help during the measurements; C. Kewish for providing the sample, which had been produced by J. Vila Comamala; V. Elser for pointing us to relevant literature; and F. Pfeiffer, M. Bech and I. Zanette for helping to improve the manuscript. This work is supported in part by a European Research Council Starting Grant, under project OptImaX (no. 279753).
The authors declare no competing financial interests.
About this article
Cite this article
Thibault, P., Menzel, A. Reconstructing state mixtures from diffraction measurements. Nature 494, 68–71 (2013). https://doi.org/10.1038/nature11806
This article is cited by
Light: Science & Applications (2022)
Scientific Reports (2022)
Nature Nanotechnology (2022)
Light: Science & Applications (2022)