## 3 Experiments
### Longitudinal Magnetic Spatial Resolution  
In polarized neutron imaging, the magnetic modulation transfer function (mMTF) is the contrast function that depends on the number of neutron spin flips per path length in a magnetic field. It was determined as frequency-dependent fringe contrast function of neutron spin-depolarizing magnetic fields from a magnetized coil. The neutrons were spin-polarized and spin-analyzed by so-called solid-state benders (m = 2.5 and m = 3), length = 40 mm, cross section 30 mm \(\times\) 30 mm [53], the neutron detector was 4k \(\times\) 4k CCD camera. The fringe function as a 2D image of a pattern was generated by a magnetized coil  
Figure 10: Spatial resolution as a function of distance of the edge from detector \(\mathrm{l}_{d}\). Note the difference between the measured and calculated resolution, the latter using the \(1/\dot{r}^{2}\) law and when the source geometry is assumed to be a pinhole rather than a mosaic crystal. \(\phi_{geom}\) = 0.1\({}^{\circ}\) means the geometric collimation obtained with a solid-state collimator and \(\phi_{mosaic}\) = 0.045\({}^{\circ}\) the measured beam collimation due to the mosaic crystal monochromator, see also [41]. Figure 9: Fitting procedure and evaluation of the spatial resolution. The spatial resolution can be directly read from the 10% contrast at 17.4 LP/mm (appr. 57 μm). 17.4 LP/mm are the maximum frequency \(f_{max}\) that appears in the image with 10% contrast. as calculated with Equation (6) and with the Radon Transform of the magnetic field [40]. Figure 6 shows the fringe pattern of coil with a quadratic and a cylindrical shape. A quadratic shaped coil creates equidistant fringe patterns different to a cylindrical shaped one and the fringe pattern is much easier to measure. For the measurements we used a quadratic shaped coil (Al body 11 mm \(\times\) 11 mm \(\times\) 150 mm, width \(\times\) height \(\times\) length), Copper wire (\(\oslash=0.90(1)\) mm, resistance = 1.2 \(\Omega\)) and currents up to 6 Ampere. The coil was placed 100 mm in front of the spin analyzer, which itself was directly in front of the 2D detector device (a CCD camera, 2k \(\times\) 2k, pixel size 13.5 um \(\times\) 13.5 um). The neutron beam (\(\lambda\) = 0.32 nm) was collimated (\(\phi_{hor}=0.1^{\circ}\), \(\phi_{vert}=0.2^{\circ}\), \(\phi_{hor},\phi_{vert}\) are the horizontal and vertical divergences) and polarized and analyzed with Swiss Neutronics supermirror benders (m = 2.5 and m = 3, each 40 mm \(\times\) 40 mm \(\times\) 40 mm, the polarization P(\(\lambda\) = 0.32 nm) was \(>\) 96%, waver thicknesses of both supermirrors were 110 um. The spatial resolution was therefore limited to 110 um in the \(\{x,y\}\) plane and concerning the vertical divergence (\(\phi_{vert}=0.2^{\circ}\)) in the \(\{x,z\}\) plane, respectively (compare Figure 4). This geometry involved a vertical geometric smearing of 0.340 mm. For each current I[A] a radiography with polarized neutrons was recorded as seen in Figure 11, exposure time was 5400 s. From each reduced image (size = 566 \(\times\) 296 pixel \(\cong\) 15.28 mm \(\times\) 7.99 mm) the background was subtracted, and outlier data removed by a [5 \(\times\) 5] median filter. Then a 150 pixel ( 4.05 mm) broad and 296 pixel (7.99 mm) high area was selected from the image and the contrast \(C=(I_{\text{max}}-I_{\text{min}})/(I_{\text{max}}+I_{\text{min}})\) calculated as a function of fringes/7.99 mm. Figure 12 shows the result. The MTF function was fitted to an error function, yielding a vertical spatial resolution of 343 um, which agrees well with the theoretical value of 0.340 mm Figure 12). The fringe pattern at current I = 6A in Figure 11 is shown enlarged in Figure 13 which is under-laid with the corresponding line graph. It can be seen that the contrast of the fringes within the plot varies, mainly caused by small stray fields and imperfections of the neutron polarizer and the neutron analyzer, but the fringes are still well separated.