Objective In order to observe more distant and fainter objects with a better resolution and signal-to-noise ratio, larger primary mirror telescopes are required to improve the diffraction limit and increase the collected light energy. This leads to problems of manufacture, testing, transportation, and launch for monolithic primary mirrors. At present, it is hard to build a monolithic primary mirror with a diameter of 8 m or larger. The segmented primary mirror is thus adopted to address these issues. However, tip-tilt errors between segments must be eliminated to meet the requirements of the light-collecting capacity and resolution. The existing tip-tilt error detection approaches mainly include the centroid detection method, phase retrieval/phase diversity (PR/PD) method, Shack-Hartmann phase sensing method, and other methods based on interferometry. In tip-tilt error detection, the centroid detection method is usually used in the coarse stages, and the PR/PD is used to eliminate the uncertainty of the centroid detection method in the fine stages. The Shack-Hartmann phase sensing method is separately used in coarse and fine stages, which also involve special-purpose hardware, complex structure, and unstable factors. Methods In this paper, a novel method, for detecting tip-tilt errors in a large capture range with a better accuracy via phase transfer function (PTF), is proposed. A mask with a sparse subpupil configuration is set on the segments' conjugate plane and serves as the entrance pupil of the tip-tilt error detection system. Then, the optical transfer function (OTF) with separated sidelobes can be obtained by the Fourier transform of the point spread function (PSF) recorded in the charge-coupled device (CCD) of the detection system, which makes it possible to separate the information of tip-tilt errors overlapped in the PSF. By analyzing the OTF sidelobes, the relationship between the phase distribution gradient of the OTF sidelobes and tip-tilt can be derived and used to extract the tip-tilt error without the measurement uncertainty of the centroid detection method, which makes the tip-tilt error detection realized with better accuracy in a large dynamic range. Simulations and experiments are conducted to verify the correctness of the proposed method. We set up a two-segmented system as shown in Fig. 2, and the tip-tilt errors are introduced from different ranges. In the small range, we introduce the tip-tilt errors from 0 to 0. 4 lambda by the step of 0. 008 lambda. In the large range, the tip-tilt errors are introduced from 0. 4 lambda to 2. 4 lambda by the step of 0. 04 lambda. In the experiment, we verify the method on the basis of the active cophasing and aligning testbed with segmented mirrors as shown in Fig. 6. The tip-tilt errors can be obtained by calculating the differences between every two centroid positions of the images formed by the segments on the focal plane. Through this experimental platform, the tip-tilt error detection method proposed in this paper is compared with the centroid detection method to achieve correctness verification. For this purpose, the mask of the tip-tilt error detection module (TEDM) is redesigned, and the original hole D is replaced with three discrete holes, as shown in Fig. 7. We have also performed preliminary simulations of the effects caused by CCD noise and figure error on the method described in this paper. Results and Discussions Simulation results show that the tip-tilt error can be detected with high accuracy over a large dynamic range as shown in Fig. 4 and Fig. 5, and the root-mean-square (RMS) has the order of magnitude of 10-15., which conforms to the detection requirements of the tip-tilt errors. Compared with the existing methods, this method does not need to divide the error detection into two stages and can effectively eliminate the measurement uncertainty of the center-of-mass detection. On the active cophasing and aligning testbed with segmented mirrors we set up before, experiments have been carried out to verify the feasibility of the method, and the RMS of detection accuracy of the method is 2. 99 x 10(-3)lambda, which meets the cophasing requirement of segmented telescopes. The experiment results are given in Table 1, Table 2, and Table 3. In addition, some factors affecting the detection accuracy of the proposed method, such as CCD noise and figure error of the tested segments, are analyzed by simulations, and the results in Table 4 and Table 5 show that in order to meet the cophasing requirement of lambda/40 (RMS), the signal-to-noise of CCD and the figure error of segments should be better than 40 dB and 0. 05 lambda (RMS), respectively. Conclusions Because of the setting of the sparse subpupil configuration and the intervention of the Fourier transform, the method in this paper effectively separates the tip-tilt errors of the segmented system in the spatial frequency domain. Then, the uncertainty of the centroid detection method during the measurement of the small errors is eliminated. The detection accuracy of the tip-tilt errors is ensured and improved. The tip- tilt error detection method simplifies the detection process and eases the demanding hardware required in existing sensing methods, and cophasing is no longer divided into coarse and fine stages that involve separate dedicated hardware solutions. This method can be adapted to any segmented primary mirror and sparse-aperture telescope system with any shape of the sub-mirror.
