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Reconstructing state mixtures from diffraction measurements

Abstract

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.

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Figure 1: Decoherence in scattering experiments.
Figure 2: Ptychography with partly coherent X-rays.
Figure 3: Reconstruction from the low-coherence X-ray experiment.
Figure 4: Imaging a simulated Ising model.

References

  1. 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)

    Article  ADS  CAS  Google Scholar 

  2. Eisebitt, S. et al. Lensless imaging of magnetic nanostructures by X-ray spectro-holography. Nature 432, 885–888 (2004)

    Article  ADS  CAS  Google Scholar 

  3. Abbey, B. et al. Keyhole coherent diffractive imaging. Nature Phys. 4, 394–398 (2008)

    Article  ADS  CAS  Google Scholar 

  4. Mandel, L. & Wolf, E. Coherence properties of optical fields. Rev. Mod. Phys. 37, 231–287 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  5. Goodman, J. W. Statistical Optics (Wiley, 2000)

    Google Scholar 

  6. Sutton, M. A review of X-ray intensity fluctuation spectroscopy. C. R. Phys. 9, 657–667 (2008)

    Article  ADS  CAS  Google Scholar 

  7. Thibault, P. et al. High-resolution scanning X-ray diffraction microscopy. Science 321, 379–382 (2008)

    Article  ADS  CAS  Google Scholar 

  8. Maiden, A. M. & Rodenburg, J. M. An improved ptychographical phase retrieval algorithm for diffractive imaging. Ultramicroscopy 109, 1256–1262 (2009)

    Article  CAS  Google Scholar 

  9. Clark, J. N. & Peele, A. G. Simultaneous sample and spatial coherence characterisation using diffractive imaging. Appl. Phys. Lett. 99, 154103 (2011)

    Article  ADS  Google Scholar 

  10. Dierolf, M. et al. Ptychographic X-ray computed tomography at the nanoscale. Nature 467, 436–439 (2010)

    Article  ADS  CAS  Google Scholar 

  11. Putkunz, C. et al. Atom-scale ptychographic electron diffractive imaging of boron nitride cones. Phys. Rev. Lett. 108, 1–4 (2012)

    Article  Google Scholar 

  12. Maiden, A. M., Rodenburg, J. M. & Humphry, M. J. Optical ptychography: a practical implementation with useful resolution. Opt. Lett. 35, 2585–2587 (2010)

    Article  ADS  Google Scholar 

  13. von Neumann, J. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl. 245–272 (1927)

  14. 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)

    Article  ADS  Google Scholar 

  15. Whitehead, L. W. et al. Diffractive imaging using partially coherent X rays. Phys. Rev. Lett. 103, 243902 (2009)

    Article  ADS  CAS  Google Scholar 

  16. Abbey, B. et al. Lensless imaging using broadband X-ray sources. Nature Photon. 5, 420–424 (2011)

    Article  ADS  CAS  Google Scholar 

  17. Candès, E. J. & Recht, B. Exact matrix completion via convex optimization. Found. Comput. Math. 9, 717–772 (2009)

    Article  MathSciNet  Google Scholar 

  18. Gross, D. Recovering low-rank matrices from few coefficients in any basis. IEEE Trans. Inf. Theory 57, 1548–1566 (2011)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  ADS  Google Scholar 

  20. Chapman, H. N. Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution: signal processing. Scanning Microsc. 11, 67–80 (1997)

    Google Scholar 

  21. Raymer, M. G. Measuring the quantum mechanical wave function. Contemp. Phys. 38, 343–355 (1997)

    Article  ADS  CAS  Google Scholar 

  22. Fienup, J. R. Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758–2769 (1982)

    Article  ADS  CAS  Google Scholar 

  23. Elser, V. Phase retrieval by iterated projections. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 20, 40–55 (2003)

    Article  ADS  Google Scholar 

  24. Guizar-Sicairos, M. & Fienup, J. R. Phase retrieval with transverse translation diversity: a nonlinear optimization approach. Opt. Express 16, 7264–7278 (2008)

    Article  ADS  Google Scholar 

  25. Thibault, P. & Guizar-Sicairos, M. Maximum-likelihood refinement for coherent diffractive imaging. N. J. Phys. 14, 063004 (2012)

    Article  Google Scholar 

  26. Pfeiffer, F. et al. Shearing interferometer for quantifying the coherence of hard X-ray beams. Phys. Rev. Lett. 94, 164801 (2005)

    Article  ADS  CAS  Google Scholar 

  27. Cerbino, R. et al. X-ray-scattering information obtained from near-field speckle. Nature Phys. 4, 238–243 (2008)

    Article  ADS  CAS  Google Scholar 

  28. Clark, J. N. et al. Dynamic sample imaging in coherent diffractive imaging. Opt. Lett. 36, 1954–1956 (2011)

    Article  ADS  Google Scholar 

  29. Cowley, J. M. Image contrast in a transmission scanning electron microscope. Appl. Phys. Lett. 58, 58–59 (1969)

    Article  ADS  Google Scholar 

  30. 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)

    Article  ADS  CAS  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. 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)

    Article  ADS  CAS  Google Scholar 

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Acknowledgements

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).

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Contributions

A.M. and P.T. designed and conducted the experiment. P.T. analysed the data and prepared the simulations. Both authors worked together to refine the methods, interpret results, write the manuscript and create the figures.

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Correspondence to Pierre Thibault.

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The authors declare no competing financial interests.

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Thibault, P., Menzel, A. Reconstructing state mixtures from diffraction measurements. Nature 494, 68–71 (2013). https://doi.org/10.1038/nature11806

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