# Digital Holographic Positioning Sensor for a Small Deployable Space Telescope
## 1 Introduction  
The demand for higher resolution and sensitivity of any optical system unavoidably leads to an increase of its aperture size. This pathway has obvious technological limitations, but for certain classes of optical systems they can be overcome. In particular, for reflective telescopes, constructing the primary mirror out of segments rather than from a monolithic piece of glass can drastically reduce the mass of the mirror and its material costs, thereby making possible the construction of ultraviolet, visible, and infrared telescopes with very large diameters. However, segmentation also introduces several complications involving the fabrication of off-axis optics, novel diffraction effects, active segment control systems, in-situ aberration correction, and the optical alignment of a large number of degrees of freedom [1].  
The problem of co-alignment of segments of a primary mirror arose for the first segmented mirrors at 10-m class ground-based astronomical telescopes [2]. In that case, the problem was solved by means of wave-front-sensing systems, such as the Shack-Hartmann [3], the pyramid [4], and the Zernike phasing sensors [5]. Further development of these techniques in the context of ground-based instrumentation is needed by creating the next generation 40-m class telescopes [6,7]. The concept of a segmented mirror is also applicable for space telescopes. In that case, the mutual alignment of the mirrors becomes a more challenging task, since the entire telescope structure is deployable. The in-space alignment technology was one of the cornerstones for the James Webb Space Telescope [8,9]and it is expected to be a driver for the next-generation flagship missions [10]. Image-based wavefront sensing techniques as the differential optical transfer function method are used in these cases. However, the application of segmented mirrors and the related automated alignment techniques is not limited to large telescopes. It was recently demonstrated that a CubeSat-class telescope can benefit in collecting area by using a small deployable primary mirror [11; 12]. In that latter case, a focal sharpening algorithm was used.  
Deployment of a CubeSat-based telescope represents an engineering task with a number of limitations. In particular, the volume and mass of the entire system are strictly limited, so the alignment system should use as few additional components as possible. Moreover, in contrast with large astronomical telescopes, the acquired image may not be suitable for calibration. While a star represents a perfect point source, most CubeSats observe the Earth surface, where the image that covers the entire field of view may have no reference features and can also be blurred by the atmosphere and the mutual motion of the satellite and the Earth.  
Thus, in the present paper, we propose a positioning sensor that can be used for alignment of a small deployable telescope. The sensor employs only onboard light sources and relies on the optics and image sensor of the telescope itself. Its operation principle is based on the comparison of a recorded interferogram with one computed for the perfect conditions and post-processing in the Fourier domain. Therefore, hereafter, we call the system a digital holographic sensor, using the general definition of digital holography [13].  
The paper is organized as follows. Section 2 describes the demonstrative optical design and performance, the primary mirror segmentation, and the deployment scheme. Section 3 presents the positioning sensor design and operation principle. Section 4 describes the interferogram post-processing algorithm and the metrics used to assess the individual components misplacements and their sensitivity. Finally, Section 5 contains the general conclusions and our proposals for future research.