A general theory of interference fringes in x-ray phase grating imaging

Abstract

Purpose: 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 twodimensional 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 Talbotdistance 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. C 2015 American Association of Physicists in Medicine.

Quantitative evaluation on 3D fetus morphology via X‐ray grating based imaging technique

“…Both gratings G1and G2, with grating pitch 10 μm and area size 25 × 25 mm, are made of tantalum with a pattern thickness of 2.1 μm and gold with a pattern thickness of 16.6 μm, respectively. The design X‐ray energy 18.8 keV of phase grating G1 can undergo a phase shift of π /2 when penetrating the grating pattern for providing the maximum fringe visibility . However, the incident X‐ray energy of 25 keV was employed in this actual measurement to secure the adequate X‐ray transmission through a water cell, in which the biological sample is immersed during the measurement.…”

“…The design X-ray energy 18.8 keV of phase grating G1 can undergo a phase shift of π/2 when penetrating the grating pattern for providing the maximum fringe visibility. 27,28 However, the incident X-ray energy of 25 keV was employed in this actual measurement to secure the adequate X-ray transmission through a water cell, in which the biological sample is immersed during the measurement. The phase shift of the phase grating G1 should be 0.376π only at 25 keV, and the fringe visibility for the third order Talbot distance will change following the equation below:…”

Sample phase gradient and fringe phase shift in dual phase grating X-ray interferometry

“…The detector just resolves the beat patterns of large periodicities, and renders other fine patterns to a constant background. Hence, different from Talbot-Lau interferometry [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], the dual phase grating interferometers directly detect interference fringes without the need of absorbing analyzer grating. This advantage brings significant radiation dose reduction as compared to Talbot-Lau interferometry.…”

Predicting visibility of interference fringes in X-ray grating interferometry

“…where p 1 is the phase grating period, λ is the x-ray wavelength, n takes odd integer values for a π-gratings, and otherwise n = 4q + 2, where q is an integer [25].…”

“…Consequently, with a broad band x-ray beam the sharp discrete Talbot distances of the self-image regime would be stretched into several downstream intervals of relatively high fringe modulations. The locations and lengths of these intervals depend on the x-ray spectrum and the grating's phase shift values [25,26]. On the other hand, the broadening of self-image regime with polychromatic x-ray offers design optimization opportunities.…”