Gauging low-dose X-ray phase-contrast imaging at a single and large propagation distance

The interactions of a beam of hard and spatio-temporally coherent X-rays with a soft-matter sample primarily induce a transverse distribution of exit phase variations δϕ (retardations or advancements in pieces of the wave front exiting the object compared to the incoming wave front) whose free-space propagation over a distance z gives rise to intensity contrast gz. For single-distance image detection and |δϕ| ≪ 1 all-order-in-z phase-intensity contrast transfer is linear in δϕ. Here we show that ideal coherence implies a decay of the (shot-)noise-to-signal ratio in gz and of the associated phase noise as z(-1/2) and z(-1), respectively. Limits on X-ray dose thus favor large values of z. We discuss how a phase-scaling symmetry, exact in the limit δϕ → 0 and dynamically unbroken up to |δϕ| ∼ 1, suggests a filtering of gz in Fourier space, preserving non-iterative quasi-linear phase retrieval for phase variations up to order unity if induced by multi-scale objects inducing phase variations δϕ of a broad spatial frequency spectrum. Such an approach continues to be applicable under an assumed phase-attenuation duality. Using synchrotron radiation, ex and in vivo microtomography on frog embryos exemplifies improved resolution compared to a conventional single-distance phase-retrieval algorithm.