# Piston Error Measurement for Segmented Telescopes with an Artificial Neural Network
## 1 Introduction  
To fulfill the demands of increasing space exploration, the aperture diameter of the telescope is getting larger and larger [1]. However, the diameter is limited by current technology to about 10 m for ground-based telescopes and a few meters for space-based telescopes because of volume and mass considerations. The segmented and deployable primary mirror was proposed to address this problem, which can effectively reduce the mass of the main mirror and the difficulty of manufacturing and transportation [2]. Ground-based telescopes that have adopted the segmented primary mirror include the already existing 10 m aperture Keck telescopes [3, 4] made of 36 segments, the 42 m diameter European Extremely Large Telescope (E-ELT) [5] made of 984 segments and the Thirty Meter Telescope (TMT) made of 492 segments with a 30 m diameter [6], which are planned in the near future. Space segmented telescope projects aim at astronomical or earth observations, such as the 6.5 m James Webb Space Telescope (JWST) [7] composed of 18 segments, to be launched very soon, and the 9.2 m Advanced Technology Large Aperture Space Telescope (ATLAST) [8] composed of 36 segments for a next-generation flagship astrophysics mission to study the universe. However, segmented-mirror technology introduced new problems. The optical path difference (OPD) between segments, which can be described by the first three Zernike polynomials (piston and tip-tilt), must be reduced to a small fraction of the wavelength to achieve a high-resolution equivalent to that of a monolithic mirror [9]. The crucial point is correcting the piston errors, which cannot be directly detected by traditionalwavefront sensors like Shack-Hartmann. The research shows that, when the piston error between segments is reduced to about \(\lambda/20\), the angle resolution is \(1.22\lambda/D\), where \(D\) is the diameter of the primary mirror and \(\lambda\) is the observation wavelength. If the piston error is larger than \(\lambda\), the angle resolution decreases to \(1.22\lambda/d\), where \(d\) is the diameter of the sub-mirror. For the JWST, which is composed of 18 1.32 m sub-mirrors, the angular resolution difference can reach five times. Thus, in order to guarantee the imaging quality of the segmented telescope, the piston error between the segments must be accurately measured, and the measurement accuracy should be 30-40 nm at least. To achieve this, the detection method of co-phasing the piston error should meet the requirements of large range and high precision. Currently, many piston error detection methods have been successfully applied to the segmented telescope. The modified Shack-Hartmann wavefront detection method has been successfully used on Keck telescopes. It measures the piston error by installing a Hartmann micro-lens array at the exit pupil. The reliable capture range of the narrow Shack-Hartmann method [10] is \(\pm\lambda/4\) while the measurement accuracy is as high as 6 nm. The broadband Shack-Hartmann algorithm [11] has a wider measurement range and generally reaches 30 \(\upmu\)m, while the measurement accuracy is around 30 nm. The quadrilateral pyramid detection method [12; 13], which utilizes a quadrangular pyramid mirror, relay lens and CCD camera, can achieve several tens of nanometers' detection accuracy in the \(\pm\lambda/2\) detection range. The dispersed fringe sensor (DFS) [14; 15] is a new piston error detection technique proposed for the next-generation space telescope. The DFS is composed of a prism, a micro-lens array and a CCD camera, which can quickly and accurately detect the piston error from a few microns to tens of microns. All above methods need optical devices which add to the detection hardware complexity. Focal-plane wavefront sensing is an elegant solution to measure the OPD since this wavefront sensor is included in the main imaging detector, simplifying the hardware and minimizing differential paths.