# Digital Holographic Positioning Sensor for a Small Deployable Space Telescope
## 2 Opto-Mechanical Concept of the Telescope  
As a demonstrative example, we consider a telescope based on the Ritchey-Chretien design [14]. The general view of the telescope optical design is shown in Figure 1.  
The telescope has the effective focal length of 2230 mm and F-number of 10.3. The angular field of view is \(\pm\)6.3 arcmin, which converts into the linear field of \(\pm\)4.1 mm. This corresponds to a commercial image sensor \(CCD97\) by \(e2v^{TM}\)[15], which has 512 \(\times\) 512 pix  
Figure 1: Optical layout of the Ritchey–Chretien type telescope with segmented primary mirror: 1—aperture stop, 2—primary mirror segments, 3—secondary mirror, 4—image sensor. els with 16 um pitch. The primary mirror is a higher-order asphere described by the following equation:  
\[z=\frac{\rho^{2}/R}{1+\sqrt{1-(1+k)\rho^{2}/R^{2}}}+\alpha_{6}\rho^{6}+\alpha_{8 }\rho^{8}, \tag{1}\]  
where \(R=-965.798\) mm is the vertex radius of curvature, \(k=-1.023\) is the conic constant, \(\alpha_{6}=3.946\times 10^{-19}\) and \(\alpha_{8}=-1.553\times 10^{-23}\) are the high-order asphericity coefficients, \(z\), mm is the surface sag, and \(\rho\), mm is the radial coordinate on the surface. The aperture stop diameter is 216.2 mm and the primary mirror is split into three hexagonal segments comprising altogether the collecting area of 174.6 cm\({}^{2}\). The convex secondary mirror has a hyperbolic surface with \(R=260.181\) mm and \(k=-2.640\) and a diameter of 48 mm. Figure 2 shows the distribution of the root mean square (RMS) spot radius of the geometric spot across the linear field of view. It is clear that the radius does not exceed 4.3 um, while the equivalent Airy disc radius is 6.9 um at 550 nm for a circumscribed aperture stop of 217.2 mm in diameter. The actual pattern of diffraction on the aperture is wider and more complex as we show further. Thus, the telescope provides a diffraction-limited image quality for the entire field of view. For comparison we consider a similar telescope with a monolithic 95 mm-diameter mirror fitting inside 1U, which represents a conventional CubeSat unit of 1 L in volume. Overlapped schematics of the two cases in comparison with the 1U cube side are shown in Figure 3A. The two telescopes have an identical F-number and an angular field. The monolithic mirror has a collecting area equal to 70.2 cm\({}^{2}\), so the gain factor is 2.49. We also compute the point spread function (PSF) for each telescope in the center of the field of view. The PSF's fr, the deployable and monolithic cases, are presented in Figure 3B,C, respectively. The corresponding angular sizes measured at the full-width at the half of the maximum (FWHM) are 0.65 and 1.33 arcsec. The complex aperture shape causes additional losses due to the diffraction on the segments edges. However, this effects decreases the Strehl ratio by only 0.9%. This telescope should fit in a stowed state inside a 1 L cube. A possible mechanical implementation of the deployment process is as follows. Figure 4A shows the telescope in stowed position. The entire telescope structure slides out of the CubeSat 1U housing (I). Then the primary mirror segments unfold (II). The secondary mirror is 376 mm away from the primary, so it is mounted on three telescopic rods, which expand (III) and then turn (IV) to set the mirror in position. Note that the detector remains static throughout all the the deployment sequence. Figure 2: RMS spot radius distribution across the linear field of view (in microns). Figure 4B depicts the telescope in a fully deployed state, where \(1\) are the primary mirror segments, \(2\) is the secondary mirror, \(3\) is the detector unit, \(4\) is the 1U CubeSat frame, \(5\) is the telescope structure basement ring with the segments tilting and rods unwrapping mechanisms, \(6\) are the secondary mirror supporting rods and \(7\) is the secondary mirror mount. All the electronics, power supply, and other systems necessary for the satellite operation may be placed in another CubeSat unit, which is not considered here.