Theoretical analysis on x-ray cylindrical grating interferometer

Grating interferometer is a state of art x-ray imaging approach, which can simultaneously acquire information of x-ray attenuation, phase shift, and small angle scattering. This approach is very sensitive to micro-structural variation and offers superior contrast resolution for biological soft tissues. The present grating interferometer often uses flat gratings, with serious limitations in the field of view and the flux of photons. The use of curved gratings allows perpendicular incidence of x-rays on the gratings, and gives higher visibility over a larger field of view than a conventional interferometer with flat gratings. In the study, we present a rigorous theoretical analysis of the self-imaging of curved transmission gratings based on Rayleigh-Sommerfeld diffraction. Numerical simulations have demonstrated the self-imaging phenomenon of cylindrical grating interferometer. The theoretical results are in agreement with the results of numerical simulations.

💡 Research Summary

The paper presents a comprehensive theoretical study of a cylindrical‑grating X‑ray interferometer, focusing on the self‑imaging (Talbot) effect that underlies phase‑contrast and dark‑field imaging. Conventional interferometers employ flat transmission gratings, which suffer from limited field‑of‑view (FOV) and reduced photon flux because the X‑ray beam strikes the gratings at an oblique angle. By curving the gratings into a cylindrical shape, the incident beam can be made normal to the grating surface across a wide angular range, thereby preserving flux and extending the usable FOV.

The authors start from the Rayleigh‑Sommerfeld diffraction integral and reformulate it in cylindrical coordinates (radius r, azimuth θ, propagation z). The grating is characterized by its radius R₀, period p, and duty cycle η. Assuming a monochromatic plane wave of wavelength λ, the complex field behind the grating is expressed as an integral over the grating transmission function T(θ′). By applying the paraxial (Fresnel) approximation and expanding T(θ′) into a Fourier series, they derive an analytical condition for the formation of self‑images at a distance Δz from the grating surface.

In a flat‑grating system the Talbot distance is simply z_T = 2p²/λ. For a cylindrical grating the curvature introduces an additional angular term, leading to a modified Talbot distance:

 z_T(θ) ≈ (2p²/λ) ·