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Reconstructing state mixtures from diffraction measurements


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.


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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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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).

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