The projection approximation versus an exact solution for X-ray phase contrast imaging, with a plane wave scattered by a dielectric cylinder

We show that exact solutions for an X-ray plane wave scattered by a dielectric cylinder with radius large compared to the radiation wavelength, give almost the same intensity fringes as seen when using the projection approximation (PA) to simulate propagation based phase contrast imaging. The characteristic Argand plane Cornu spiral and hypocycloid from the complex wavefield beyond the cylinder was observed to agree with the PA simulations. Asymptotic approximations to the exact analytical solution simplify to the projection approximation at the centre of the cylinder. The projection approximation underestimates observed fringes at a propagation distance comparable with the cylinder radius, but converges to the exact solution at larger propagation distances. This demonstrates that the complex ray approach of the projection approximation is insufficient, in comparison to the exact approach, when the propagation distances are sufficiently small.