# Cold and Thermal Neutron Single Grating Dark-Field Imaging Extended to an Inverse Pattern Regime
## 3 Results  
First, the visibilities of the projected patterns of the gratings were measured at NEUTRA. Two separate sets of scans were carried out for two different positions of the linear stage with the grating along the beam direction, to extend the total range of grating-detector distances. Figure 2 shows plots of the visibility V\({}_{0}\) for both the line and the hexagonal pattern gratings as a function of the distance L\({}_{\mathrm{g}}\) between the grating and the imaging detector. As expected, the maximum visibility obtained within the measurements was found at the shortest grating-detector distance for both gratings. As the distance increases, the visibility decreases due to the blurring effect caused by the finite beam collimation. However, only the initial slope follows the expected blurring effect according to pinhole imaging [10], and as the distance increases, the visibility starts to rise again reaching a second relative maximum. In this regime, an inverse pattern of the original grating pattern with the same period is observed (see Figure 2b). As the distance increases further, a second minimum is found, before the visibility starts to rise again and the regular grating pattern is recovered at a third maximum. It can be observed that the line pattern yields an overall higher visibility than the hexagonal pattern due to the blurring impact in only one dimension and larger grating period, which is better resolved by the detector system. Furthermore, the larger period of the line grating compared to the hexagonal grating results in a slower degradation of the pattern and hence a longer distance before pattern inversion sets in. These results are all in agreement with simple linear optics ray tracing simulations and, thus, do not involve any phase or interference effects. In the second experiment, which was performed at the ICON beamline, the attenuation and dark-field signals of a stack of four 300 \(\upmu\)m thick electrical steel sheet samples were  
Figure 2: **(a)** Modulation of the grating visibility as a function of the distance between the grating and the neutron detector. The green lines with cross markers correspond to the line pattern grating whereas the blue lines with star markers correspond to the hexagonal 2D pattern grating. The dark and light lines indicate two separate grating distance scans acquired at two different manually fixed positions on an optical bench of the motorized grating linear stage. **(b)** Projections of the hexagonal and line pattern gratings obtained at the grating distances indicated with the black arrows in **(a)**. obtained. Here, we succeeded in accessing autocorrelation lengths in the extended range from 13.1 and 253.4 nm, which was achieved by exploiting both the regular and the inverse pattern produced by the hexagonal grating, measuring, respectively, at grating distances Lg 145 and 275 mm. Figure 3 shows attenuation and dark-field contrast images of the sample as well as their average value as a function of the probed autocorrelation lengths. As expected, the attenuation value does not significantly change with the probed autocorrelation lengths regardless of the type of pattern. However, for short autocorrelation lengths probed, the attenuation curve appears to suffer slightly from bias due to scattered neutrons that are typically found for short sample to detector distances [21]. As the distance between the sample and the detector increases, the bias dissipates. This is because the range of scattering angles, for which the scattered neutrons do not escape the detector and contribute to the measured beam intensity, decreases with distance. Instrumental and software corrections to this bias have been introduced in [22; 23]; however, compatibility with DF imaging is not yet established. The dark-field contrast as a function of probed autocorrelation length shows the typical monotonically decreasing trend, as reported in the literature for similar samples [13; 24].