## 4 Summary  
In this paper, at first some basics on spatial resolution and modulation transfer function determination are given and it is proposed to obtain the modulation transfer function from edge image experiments. The edge data are fit to the convolution of a Gauss function with a step function, resulting in a smooth function for the edge image (and for the Gauss function) the Fourier transform is then applied to the (already smoothed) Gauss function, yielding the corresponding modulation transfer function and thus the spatial resolution at 10% contrast. In this setup (PONTO II) the spatial resolution as a function of the distance of the edge from the detector differs significantly from the pin hole--source geometry. The use of a mosaic monochromator increased the L/D ratio by a factor of 2.2 from L/D(pin hole) = 573 to L/D (mosaic monochromator) = 1270, which has already been demonstrated with another instrument [41]. The spatial resolution for unpolarized neutrons at PONTO II was determined to be 57(3) \(\upmu\)m for an object at a distance of 5 mm from the detector, which is an improvement from earlier measurements (80 \(\upmu\)m) [35]. When determining spatial resolution in the case of polarized neutron imaging, it is necessary to distinguish longitudinal and lateral resolution. The spin depolarization was measured using a special neutron supermirror device (SwissNeutronics) as polarizer and analyzer elements. The measurement is governed by the Larmor precession \(\omega_{L}\) resp. by the spin rotation angle \(\phi\) (see Equation (6)). The longitudinal resolution was determined by increasing the number of spin flips (fringes) within a given path length. The measured spatial resolution by the modulation transfer function (345(20) \(\upmu\)m) agrees well with the calculated value (350 \(\upmu\)m) and with the \begin{table}
\begin{tabular}{c c c} \hline
**Shift Nr.** & **Spin-Flip Distance [\(\upmu\)m]** & **Standard Dev. [\(\upmu\)m]** \\ \hline
0 & 396 & 87 \\
10 & 381 & 51 \\
25 & 381 & 46 \\
30 & 380 & 48 \\
50 & 415 & 73 \\ \hline \end{tabular}
\end{table} 
Table 1: Fringe pattern. Figure 16: Line plot from image 10 and 25, mean distance maximum-maximum = 381(46) \(\upmu\)m, see also Table 1.  
calculated convolution value (376 \(\upmu\)m). The longitudinal resolution refers to the largest B in the sample, the smallest B results from the minimum spin rotation which can still be determined. In addition, the magnetic resolution depends on the incident beam polarization P, and both the largest and smallest B to be detected also depend strongly on the neutron wavelength used. Lateral spatial resolution (\(\Delta\)z) was measured by moving a given fringe pattern, generated with a magnetized square coil, across the 2D detector in steps of 50 \(\upmu\)m. The detector pixels are a known parameter, as is the fringe pattern, so experimental convolution yields the correct values (380(48) \(\upmu\)m for a spin-flip. Due to the vertical beam divergence \(\phi_{vert}=2\)\(\phi_{horiz}\), \(\Delta x\) can be given as \(\frac{1}{2}\Delta z=190\)\(\upmu\)m.  
Finally, it can be stated that the wave nature of neutrons does not affect spatial resolution in magnetic field imaging as described in this paper. The wave nature would cause phase effects, i.e., refraction and diffraction (interference effects). Both interactions require sharp phase transitions [42; 43; 44; 45; 46], or ordered structures, for example, in bodies with magnetic domains [54] or in magnetic structures [55], which are not discussed here. Conceptualization, W.T. ; methodology, W.T. ; experiments, R.K.; validation, W.T., R.K.; writing--review and editing, W.T., R.K. Both authors have read and agreed to the published version of the manuscript. This research was funded by the Federal Ministry of Education and Research - BMBF, project 05 K10 FK1. Not applicable.