title: A general theory of interference fringes in x-ray phase grating imaging.
abstract: OBJECTIVE
  The authors note that the concept of the Talbot self-image distance in x-ray phase grating interferometry is indeed not well defined for polychromatic x-rays, because both the grating phase shift and the fractional Talbot distances are all x-ray wavelength-dependent. For x-ray interferometry optimization, there is a need for a quantitative theory that is able to predict if a good intensity modulation is attainable at a given grating-to-detector distance. In this work, the authors set out to meet this need.
  
  
  METHODS
  In order to apply Fourier analysis directly to the intensity fringe patterns of two-dimensional and one-dimensional phase grating interferometers, the authors start their derivation from a general phase space theory of x-ray phase-contrast imaging. Unlike previous Fourier analyses, the authors evolved the Wigner distribution to obtain closed-form expressions of the Fourier coefficients of the intensity fringes for any grating-to-detector distance, even if it is not a fractional Talbot distance.
  
  
  RESULTS
  The developed theory determines the visibility of any diffraction order as a function of the grating-to-detector distance, the phase shift of the grating, and the x-ray spectrum. The authors demonstrate that the visibilities of diffraction orders can serve as the indicators of the underlying interference intensity modulation. Applying the theory to the conventional and inverse geometry configurations of single-grating interferometers, the authors demonstrated that the proposed theory provides a quantitative tool for the grating interferometer optimization with or without the Talbot-distance constraints.
  
  
  CONCLUSIONS
  In this work, the authors developed a novel theory of the interference intensity fringes in phase grating x-ray interferometry. This theory provides a quantitative tool in design optimization of phase grating x-ray interferometers.
authors.affiliation: Department of Radiology, University of Alabama at Birmingham, Birmingham, Alabama 35249.
authors.forename: Aimin
authors.initials: A
authors.lastname: Yan
authors.affiliation: Department of Radiology, University of Alabama at Birmingham, Birmingham, Alabama 35249.
authors.forename: Xizeng
authors.initials: X
authors.lastname: Wu
authors.affiliation: Center for Bioengineering and School of Electrical and Computer Engineering, University of Oklahoma, Norman, Oklahoma 73019.
authors.forename: Hong
authors.initials: H
authors.lastname: Liu
mesh_terms: D005583:Fourier Analysis; D007091:Image Processing, Computer-Assisted / Q000379:methods*; D007368:Interferometry; D008962:Models, Theoretical; D014057:Tomography, X-Ray Computed / Q000379:methods*; D014961:X-Ray Diffraction