Discontinuities in the wavefront phase or amplitude affect curvature sensor (CS) response. These discontinuities may be the result of an improper alignment of the elements of a segmented mirror in either piston or tip-tilt. We describe the CS response with an analytical approach based on the Fresnel approximation of the diffracted field and show that a CS is capable of detecting misalignment among segments. The response model that we present leads to a fast algorithm for the measurement of segment positions. As an application, we focus on piston co-phasing. The paper concludes with the characterization of the integrated curvature signal, which simplifies the analysis of the CS response to misalignments. The low processing time and principal memory requirements of the algorithm make it suitable for extremely large telescopes (ELTs).
KEYWORDS: Mirrors, Error analysis, Sensors, Image segmentation, Integrated modeling, Monte Carlo methods, Diffraction, Segmented mirrors, Telescopes, Chemical elements
In this paper, we study the ability of a curvature sensor (CS) to measure co-phasing errors in a segmented primary mirror. We study a new fast and low memory-consuming algorithm based on the definition of an integrated curvature signal. This signal results from the integration of the CS response in enclosures matching the shapes of the segments in the mirror. This gives a low number of measurements, equal to the number of segments. The determination of the position of the segments is speeded with an analytical response model, which we formulate in matrix form. In this way, we can make an analysis of the error propagation that is valid for measurement errors or imprecision of the response model. The procedure is well suited to mirrors with a large number of segments, as in the case of ELTs.
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