# Cold and Thermal Neutron Single Grating Dark-Field Imaging Extended to an Inverse Pattern Regime
## 2 Materials and Methods  
The measurements were carried out at the NEUTRA and ICON beamlines at the SINQ facility (PSI, Villigen, Switzerland). NEUTRA has a thermal spectrum with a mean neutron wavelength of 1.5 A and maximum at 1.1 A [17], whereas ICON has a cold spectrum with a mean neutron wavelength of 2.8 A and maximum at 1.5 A [18]. A 'white', polychromatic beam without monochromatization was used for all measurements presented here. For both beamlines, the neutron detector was comprised of a 200 \(\upmu\)m thick scintillation screen (LiF, ZnS(Ag)), and an optical Andor's iKon-L CCD camera (2048 \(\times\) 2048 pixels). A 100 mm Zeiss f/2 objective lens was used with a magnification such to achieve an effective pixel size of 35 \(\upmu\)m, leading to a field of view of 72 \(\times\) 72 mm\({}^{2}\). Figure 1 shows the schematic instrumental setup used in this work. The spatial collimation of the beam, and consequently the spatial resolution was mainly limited by the front square aperture of 20 \(\times\) 20 mm\({}^{2}\) that was placed at a distance of approximately 7 m upstream of the scintillation screen at both beamlines. An absorption grating, placed at a distance L\({}_{\text{g}}\) from the imaging detector, produced the spatial modulation of the neutron beam. It consisted of a 70 \(\times\) 70 mm\({}^{2}\) absorbing pattern of 20 \(\upmu\)m thick gadolinium deposited on a 300 \(\upmu\)m thick round silicon wafer. The grating pattern was produced by laser ablation of the initially fully gadolinium-coated wafer. The two types of patterns used for the measurements presented here were parallel lines with a width of 182.5 \(\upmu\)m and a period \(p=365\)\(\upmu\)m, and round holes with a diameter of 175 \(\upmu\)m distributed on a hexagonal grid with a period equal to twice the hole diameter in the three symmetry directions (see Figure 2). While measurements with the line pattern are only sensitive to scattering perpendicular to the line direction, the hexagonal pattern allows probing the scattering signal in the three symmetry directions of the pattern with a spacing of 60\({}^{\circ}\). Furthermore, using directional signal processing routines, such as the ones presented in [11; 19; 20], the scattering signal in the three individual directions can be separated. The absorption grating was mounted on a linear stage, which allowed for a motorized movement of the grating up to 135 mm in the direction parallel to the neutron beam. In addition, the linear stage could be manually moved on an optical bench. The sample was placed downstream of the grating at a distance L\({}_{\text{s}}\) upstream of the imaging detector. The sample stage featured three linear stages that enabled motorized positioning of the sample in the three principal orthogonal directions. The movement of the sample along the beam direction with a 145 mm range was required for the autocorrelation length scans. For the processing of the images \(I\) and \(I_{0}\) acquired, respectively, with and without the sample, an analyzing window \(\mathcal{A}(x,y)=I([x-w,x+w],[y-w,y+w])\), where \((2w+1)\) is the window size along one axis and \(x\) and \(y\) are the image pixel coordinates, was introduced. For each pixel \((x,y)\) the attenuation A was calculated as  
\[\text{A}(x,y)=\frac{<\mathcal{A}(x,y)>}{<\mathcal{A}_{0}(x,y)>}, \tag{1}\]  
where \(<>\) denotes the average operator applied to the respective analyzing window. This was performed instead of a simple \(I/I_{0}\) normalization to reduce artifacts caused by drifts  
Figure 1: Sketch of the instrumental setup. in the grating position, which may happen during sample exchanges or due to system perturbations. Analogously, the visibility V was calculated as  
\[\mathrm{V}(x,y)=\frac{\sigma(\mathcal{A}(x,y))}{<\mathcal{A}(x,y)>}, \tag{2}\]  
where \(\sigma()\) denotes the standard deviation operator. Finally, the dark-field contrast DF was calculated as the ratio of the visibilities with and without the sample \(\mathrm{DF}(x,y)=\mathrm{V}(x,y)/\mathrm{V}_{0}(x,y)\).