Directional dark-field implicit x-ray speckle tracking using an anisotropic-diffusion Fokker-Planck equation

Konstantin M. Pavlov, David M. Paganin, Kaye S. Morgan, Heyang Li, Sebastien Berujon, Laurène Quénot, Emmanuel Brun

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12 Citations (Scopus)


When a macroscopic-sized noncrystalline sample is illuminated using coherent x-ray radiation, a bifurcation of photon energy flow may occur. The coarse-grained complex refractive index of the sample may be considered to attenuate and refract the incident coherent beam, leading to a coherent component of the transmitted beam. Spatially unresolved sample microstructure, associated with the fine-grained components of the complex refractive index, introduces a diffuse component to the transmitted beam. This diffuse photon-scattering channel may be viewed in terms of position-dependent fans of ultrasmall-angle x-ray scatter. These position-dependent fans, at the exit surface of the object, may under certain circumstances be approximated as having a locally elliptical shape. By using an anisotropic-diffusion Fokker-Planck approach to model this bifurcated x-ray energy flow, we show how all three components (attenuation, refraction, and locally elliptical diffuse scatter) may be recovered. This is done via x-ray speckle tracking, in which the sample is illuminated with spatially random x-ray fields generated by coherent illumination of a spatially random membrane. The theory is developed and then successfully applied to experimental x-ray data.

Original languageEnglish
Article number053505
Number of pages13
JournalPhysical Review A
Issue number5
Publication statusPublished - Nov 2021


  • x-ray
  • phase contrast
  • Synchrotron

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