## 2 Early Development in MCT Beamline
### Spatial Resolution of the White-Beam Detector  
The two imaging modalities offered by the MCT beamline in the early stage of its operation are absorption-contrast and propagation-based phase-contrast. The point spread function (PSF) of these imaging setups can be described as a combination of the blurring from the source size due to the geometric setup and the blurring from the detector itself. Figure 4: (**a**). White-beam detector with three objective lenses. (**b**). Monochromatic-beam detector with three objective lenses. Figure 3: (**a**). Image of the “first light” captured by a visible light camera connected to a computer. (**b**). Diagnostic screen inside vacuum tube (blue dot line square) with visible light camera system (green arrow direction) that captured the X-rays (red arrow direction). The blurring from the source is due to the geometric magnification in the propagation-based imaging setup, whereas the blurring from the detector is mainly caused by the blurring associated with the scintillator thickness. A thin scintillator will contribute less to the PSF but then lowers the efficiency and so a compromise is needed. The standard deviation for the (normalized) Gaussian for the PSF for our imaging system can be presented as follows [2]:  
\[\sigma_{sys}=\sqrt{\left(\frac{M-1}{M}\right)^{2}\sigma_{s}^{2}+\frac{\sigma_{ d}^{2}}{M^{2}}} \tag{5}\]  
where \(M\) is geometric magnification, \(\sigma_{s}\) is the Gaussian standard deviation for the source size, and \(\sigma_{d}\) is the standard deviation for the line spread function (LSF) of the detector, which, in this case, is mainly due to the scintillator blurring. The geometric magnification, \(M\), is dependent on the distance between the source and the sample (\(R_{1}\)) and the sample and the detector (\(R_{2}\)) as \(M=(R_{1}+R_{2})/R_{1}\). For the true absorption-contrast mode, where \(R_{2}=0\) and \(M=1\), the first term of Equation (5) will be zero. This first term will become important when the sample to detector distance is large, \(M>1\). In the current set-up the source-sample distance \(R_{1}\) is 21.5 m with the source size, \(\sigma_{s}\), of 87 \(\upmu\)m horizontally and 58 \(\upmu\)m vertically. [PERSON] et al. [26] use the expression for the Gaussian LSF at 90% and 50%. For non-Gaussian LSFs, we can describe \(\sigma_{d}\) as follows:  
\[\sigma=\frac{\left(\sigma_{(fu90\%)}+\sigma_{(fu50\%)}\right)}{2}\]  
\[\sigma\equiv\sigma_{d}=\sqrt{\left(\frac{0.19\;n\lambda_{\wp}}{NA}\right)^{2} +\left(0.043\;s\;NA\right)^{2}}+\sqrt{\left(\frac{0.12\;n\lambda_{\wp}}{NA} \right)^{2}+\left(0.028\;s\;NA\right)^{2}} \tag{6}\]  
where \(n=1\) for the condition in air, \(NA\) is the numerical aperture, \(s\) is the scintillator thickness, and \(\lambda_{\wp}\) is the wavelength of visible light produced by the scintillator in response to the X-ray beam (550 nm in this case). For the MCT white-beam detector, we will have \(\sigma_{d}\) as shown in Table 1 as calculated using Equation (6). The information regarding the numerical aperture, scintillator thickness, FOV, and effective pixel size are also presented in Table 1. The white-beam detector uses LuAG:Ce scintillators with a different thickness for each objective lens of \(1\times\), \(5\times\), and \(20\times\), respectively. The corresponding system standard deviation, \(\sigma_{sys}\), for several sample-to-detector distances of 160 mm, 700 mm, and 1410 mm can be found in Table 2, as calculated using Equation (5), for the horizontal direction. The horizontal source blurring was calculated using the horizontal source size of 87 \(\upmu\)m. Similarly, the vertical source blurring (not presented in the table) can be calculated using the vertical source size of 58 \(\upmu\)m. The corresponding full width at half maximum (FWHM) can be calculated as \(\mathrm{res_{FWHM}}=2\sqrt{2\;ln\;2}\;\sigma_{sys}\approx 2.355\;\sigma_{sys}\).