# Digital Holographic Positioning Sensor for a Small Deployable Space Telescope
## 3 The Positioning Sensor Design  
The position of each optical element can be defined from an interference pattern formed at the telescope focal plane. Then the measurement reduces to the definition of a few sources position from a matrix of counts corresponding to the interference result, which represents an over-determined mathematical problem and requires simplification of the main equations and the use of numerical methods. A similar problem was considered before in the field of radio engineering [16]. To measure the misalignment of movable optical components we propose to mount a laser point source in the outer-middle side of each primary mirror segment as shown in Figure 5A. Practically, they may represent ends of fiber coupled with a single laser source set up elsewhere. We assume that with such an implementation, the illuminating system will be compact and lightweight, so it would not change the telescope structure mechanical properties. Moreover, it is assumed that the size of the fiber end is negligible in comparison with the segment structure and that the connection between the fiber and the segment is perfectly rigid, so their mutual movements are not included into the sensitivity budget. A typical numerical aperture of the fiber exceeds the angular size of the detector, so a significant part of the beam will pass by the detector and, therefore, it is not shown in the figure. However, this lost fraction of the beam should be taken into account for \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Primary Mirror** & **Secondary Mirror** \\ \hline Distance after, mm & 0.1 & 0.1 \\ \hline Vertex radius, fringes & 3 & 5 \\ \hline Irregularity, nm & 6.3 & 6.3 \\ \hline X/Y decenter, mm & 0.1 & 0.04 \\ \hline Tip/Tilt, deg & 0.007 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical design parameter tolerances. the definition of the laser source, telescope mechanical design, and black coatings usage. These sources illuminate the secondary mirror and the reflected light forms an interference pattern in the telescope focal plane. A sample pattern is presented in Figure 5B. The path of each ray reaching the detector is defined by three points: the laser point source \((x_{L},y_{L},z_{L})\), the reflection point on the secondary mirror \((x_{M},y_{M},z_{M})\), and the point at detector \((x_{D},y_{D},z_{D})\). Assuming that the global coordinates origin located in the detector center, we can assign \(z_{D}=0\). Then, we can write down \(z_{L}=z_{P}+d_{1}\), where \(z_{P}\) is the primary mirror sag calculated from (1) and \(d_{1}\) is the distance between the primary mirror surface center and the detector center. Similarly, \(z_{M}=z_{S}+d_{1}+d_{2}\), where \(z_{S}\) is the secondary mirror sag calculated from (1) and \(d_{2}\) is the distance between the mirrors centers. Then the ray tracing problem is reduced to definition of the \((x_{M},y_{M})\) coordinates. We use the reflection equation in vector form:  
\[\vec{R}_{i}\times\vec{N}=-\vec{R}_{r}\times\vec{N}, \tag{2}\]  
where \(R_{i}\) and \(R_{r}\) are the direction vectors of the incident and reflected rays, respectively:  
\[\vec{R}_{i}=\frac{\overline{(x_{M}-x_{L},y_{M}-y_{L},z_{M}-z_{L})}}{|\overline {(x_{M}-x_{L},y_{M}-y_{L},z_{M}-z_{L})}|}, \tag{3}\]  
\[\vec{R}_{r}=\frac{\overline{(x_{D}-x_{M},y_{D}-y_{M},z_{D}-z_{M})}}{|\overline {(x_{D}-x_{M},y_{D}-y_{M},z_{D}-z_{M})}|}, \tag{4}\]  
and \(\vec{N}\) is the normal vector to the secondary mirror surface in the incidence point:  
\[\vec{N}=\frac{\overline{(\partial z_{s}/\partial x,\partial z_{s}/\partial y, 1)}}{|\overline{(\partial z_{s}/\partial x,\partial z_{s}/\partial y,1)}|}. \tag{5}\]  
The ray tracing Equation (2) can be solved by some numerical method, e.g., simplex minimization [17].