X-ray phase-contrast imaging

X-ray imaging is a standard tool for the non-destructive inspection of the internal structure of samples. It ﬁnds application in a vast diversity of ﬁelds: medicine, biology, many engineering disciplines, palaeontology and earth sciences are just few examples. The fundamental principle underpinning the image formation have remained the same for over a century: the X-rays traversing the sample are subjected to diﬀerent amount of absorption in diﬀerent parts of the sample. By means of phase-sensitive techniques it is possible to generate contrast also in relation to the phase shifts imparted by the sample and to extend the capabilities of X-ray imaging to those details that lack enough absorption contrast to be visualised in conventional radiography. A general overview of X-ray phase contrast imaging techniques is presented in this review, along with more recent advances in this fast evolving ﬁeld and some examples of applications.


Introduction 1
The use of X-rays for imaging the internal structure of samples quickly spread

Absorption imaging 36
The sketch in Figure 1 reports the arrangement that is typically used in ra-37 diography by using an X-ray source and an image receptor. It is a transmission- We can then calculate the corresponding intensity profile by using Equation 1. 59 In order to do so we need to specify the working energy (30 keV), the materials 60 (aluminium for the sphere and water for the embedding material) and we further 61 assume a constant incident intensity I 0 (x, y) = 1.

62
It is often the case, for example when using conventional laboratory sources 63 such as X-ray tubes, that the radiation is polychromatic and its spectrum ex-64 tends over a range of several tens of keV. This can be included in Equation 1 by response D(λ) Each monochromatic component of the X-ray beam contributes independently 69 to the contrast, with a weight that is equal to the relative probability of emis-70 sion and detection, and with the attenuation coefficient characteristic of that 71 particular energy (for example see [14]). The phase of the waves travelling through the sample contributes to the 74 modulation of the detected intensity in an X-ray phase-contrast imaging system. 75 This can be described by means of the complex refractive index [15] 76 where the decrement to unity δ governs the phase shifts while β the absorption. where r e is the classical electron radius. It is worth noting that δ is typically 81 larger than β. By taking for example water at 30 keV, we obtain δ ≈ 2.56 · 10 −7 82 and β ≈ 1.36·10 −10 . Another key difference between the two parameters is their  The phase shift imparted by the sample to the X-ray wave is given by where a single energy was used for the X-ray beam. I(x, y; M, λ) = that describes the intensity distribution a the image receptor plane from a pure 143 phase object. M = (R 1 +R 2 )/R 1 is the geometrical magnification. The contrast 144 from a pure phase object vanishes when R 2 → 0, which is the typical condition 145 for conventional (contact) radiography and the phase term is directly propor-146 tional to the propagation distance R 2 . Another feature of interest is that the 147 monochromaticity of the radiation is not essential for this type of imaging. A 148 necessary condition, however, is that the radiation must have a certain degree 149 of spatial coherence [13]: where σ s is the standard deviation of the source intensity distributon. The 151 coherence length l c has to be comparable to or larger than the inverse spatial 152 frequency of the feature of interest [35] in order to obtain signifcant phase con-153 trast. In practice this means that the source has to be relatively small or that 154 the object must be placed at a relatively large distance R 1 from it. Another re-155 quirement is that the imaging system must have spatial resolution high enough 156 to not wash out the interference fringes. This is conveniently summarised by where σ t and σ d are the standard deviations of the system's and of the detec- 159 tor's point spread function, respectively. Another point to be noted is that the 160 diffraction term 161 σ m = 1 2 λR 2 2 (13) becomes less significant for increasing X-ray energies.

162
The intensity projection image, acquired with a certain propagation distance 163 between the sample and the detector, will contain a mixture of contributions  up is sketched in Figure 6. The X-ray beam is usually wide enough to cover Because X-rays are deflected away from the beam axis (a glass sphere in air acts 215 as a diverging lens in the X-ray regime), the angle of incidence of the radiation on 216 the analyser will be changed in two opposite ways at the two edges of the sphere.

217
On one side, this will result in a higher probability of transmission through the where p 2 is the period of G 2 and a i and Φ i are the amplitude and phase coeffi- tional radiography, is given by The differential phase-contrast projection image of the sample is calculated by , and by considering where d is the distance between G 1 and G 2 . Dark-field images are obtained by 261 first computing the normalized oscillation amplitude and then by taking the ratio of this quantity, with and without sample in the which does not show changes (V (x, y) = 1) for samples with negligible or absent 265 small-angle scattering and is reduced (V (x, y) < 1) when scattering occurs.    where µ mn = µ m + µ n , σ 2 mn = σ 2 m + σ 2 n and A mn = A m A n (1/ 2πσ 2 mn ). Both