Interference pattern with an analytical solution in modified Shack-Hartmann sensor

Segmented telescope is an effective way to realize high-resolution observations in astronomy. An important work for high-resolution observations using segmented telescopes is phasing the segmented primary mirror. Modified Shack-Hartmann sensor. Is proposed for piston error detection. The interference pattern created by a circular lens placed across two adjacent mirrors in exit pupil plane is used as the signal of the modified Shack-Hartmann sensor. Piston errors need to be extracted from the interference pattern. The offset of lens and gap error of adjacent mirrors causes the distortion of interference pattern, and leads to a reduction in the detection accuracy of existing piston error extraction techniques. In this paper, we propose to replace the circular lens with a square lens and the mathematical model of the corresponding interference pattern is modeled by Fourier optics, including the one-dimensional and two-dimensional analytical solution of the interference pattern. The simulation results show that the proposed analytical solution can effectively characterize the interference pattern in the ideal situation and in the presence of lateral offset of the lens and the gap error of the adjacent mirrors situation. The results presented here give a deeper insight into the interference pattern of modified Shack-Hartmann sensor, and are of great help for developing new piston error detection techniques based on modified Shack-Hartmann sensor.