# A Single Motor-Driven Focusing Mechanism with Flexure Hinges for Small Satellite Optical Systems
## 2 Materials and Methods  
### Definition of Mis-Alignments of the Schmidt-Cassegrain Optical System  
Undesired mis-alignments cause the degradation of image resolution, which consists of independent parameters in three-dimensional cartesian space (i.e., De-space, De-center, and Tilt), as shown in Figure 1. De-space represents the undesired deformation of the secondary mirror that occurs parallel to the direction of the optical axis. De-center indicates an undesired lateral displacement perpendicular to the optical axis, and Tilt indicates an irregular angle or skews relative to the optical axis, measured as rotation about axes perpendicular to the optical axis. \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Description** \\ \hline Optics Type & Schmidt-Cassegrain \\ Schmidt plate diameter & 200 mm \\ Primary mirror diameter & 200 mm \\ Primary mirror curvature (radius) & 551.71 mm \\ Secondary mirror diameter & 65 mm \\ Secondary mirror curvature (radius) & 185.92 mm \\ Altitude & 685 km \\ GSD & 2.8 m \\ Detector & 35.8 \(\times\) 23.8 mm CMOS sensor \\ Pixel size & 8.2 \(\upmu\)m \\ Swath Width & 12 km \\ MTF & \(\geq\)30\%, @Optical Design, Nyquist \\  & Freq. \\ Linear Obscuration & \(\leq\)0.35 \\ F/\# & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Conceptual optical system specifications. Approximation of MTF Performance  
The MTF is a function of spatial frequency, which correlates the resolution and the contrast for the image. The variation of MTF occurs due to the mis-alignments of optics. Indeed, these parameters cause diffraction of the light when it forms an image; thereby, image's quality and/or resolution (pixel) could be degraded. Among the mis-alignment parameters, the ranges of De-space and Tilt are crucial because variations in these values could degrade MTF. Therefore, we limited certain ranges for the De-space and the Tilt, and defined the De-center to correspond to a single pixel size of the conceptual detector (8.2 \(\upmu\)m). Here, once the De-center is within a range of 8.2 \(\upmu\)m, the variation of MTF does not exceed the requirements that we assigned in Table 1. To verify these, in this section, the variation in MTF values was demonstrated by using an optics simulation program (CODE-V). Here, we employed the specific frequency--Nyquist frequency corresponds to the pixel size of the sensor. Since the pixel size of the employed image sensor is 8.2 \(\upmu\)m, the Nyquist frequency is 61.0 lp/mm. With this condition, a comparison group was made to figure out their proper ranges, as shown in Table 2. Each case corresponds to a variation in Tilt, ranging from 0 to 200 \(\upmu\)rad, with steps of 100 \(\upmu\)rad. Here, two axes are defined as the \(x\) and \(y\) axes, respectively. The Tilts that occur for each axis are defined as \(\alpha\) and \(\beta\)-Tilts, respectively. When \(\alpha\) and \(\beta\)-Tilts were 200 \(\upmu\)rad (Case 3), respectively, the MTF was reduced by 5%, compared to Case 1 (without Tilts), as shown in Figure 2. On the other hand, once the De-space was over \(\pm\)5 \(\upmu\)m, the MTF degraded to below 30%, where the MTF value was a minimum. Furthermore, the degradation of MTF due to De-space is more significant than the degradation caused by Tilt. Therefore, the De-space and \(\alpha\) and \(\beta\)-Tilts should be limited by \(\pm\)5 \(\upmu\)m and 200 \(\upmu\)rad, respectively. Also, the available range of \begin{table}
\begin{tabular}{l c c c} \hline \hline  & \(\alpha\)**-Tilt (\(\upmu\)rad)** & \(\beta\)**-Tilt (\(\upmu\)rad)** & **MTF at De-Space = 0** \\ \hline CASE 1 & 0 & 0 & 0.398 \\ CASE 2 & 100 & 100 & 0.395 \\ CASE 3 & 200 & 200 & 0.385 \\ CASE 4 & 100 & 0 & 0.396 \\ CASE 5 & 200 & 0 & 0.391 \\ CASE 6 & 0 & 100 & 0.396 \\ CASE 7 & 0 & 200 & 0.391 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Case study to identify the variation of MTF. Figure 1: Undesired mis-alignments that occur at the secondary mirror; (**a**) De-space, (**b**) De-center, (**c**) Tilt, and (**d**) definition of \(\alpha\)-Tilt and \(\beta\)-Tilt.