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# Decoupling induced aberrations from surface error

The measurement repeatability of subapertures is the premise of stitching interferometry, which is based on the overlapping consistency. Stitching optimization cannot cancel the influence of noise, but we can try to depress it by carefully designing the subaperture layout. In our experience of simulations and experiments, the uncertainty level of stitching is comparable to the noise level in subaperture measurements.40,50 Stitching optimization is responsible for separating the misalignment-induced aberrations as well as systematic errors from subaperture measurements, but some components of the surface error may physically appear, like a systematic error in subapertures, and therefore be incorrectly self- calibrated out. For example, rotationally symmetric spherical aberrations produce an identical map for subapertures with the same amount of off-axis distance. This problem can be avoided when more than one ring of subapertures is designed on the full aperture with different off-axis distances. Systematic errors, including the TS reference error and null optics error, are identical for all subapertures, whereas the surface error can only be identical for those lying on the same ring. It applies to null or non-null subaperture stitching with or without null optics. Self-calibration of systematic error will be further discussed and demonstrated in the following case studies.

For near-null subaperture stitching where the near-null optics is required to be reconfigured to generate varying aberrations, identical or systematic errors are produced only for those subapertures lying on the same ring. However, the identical error produced by a rotationally symmetric surface error is not inherently the same as that produced by the near-null optics. The former contributes nothing to the overlapping inconsistency, whereas near-null optics can introduce error that increases the overlapping inconsistency. As QR decomposition yields the LS solution with a minimal norm during stitching optimization, no extra error will be added to the surface error even if it does not contribute to the objective function of overlapping inconsistency. On the other hand, those components that have no effect on the inconsistency induced by near-null optics are still coupled in the surface error. Therefore, how the systematic error behaves in stitching depends on its composition, and reasonable planning of subaperture measurements can be helpful for avoiding error accumulation. For instance, rotation of the test surface is preferred to avoid accumulation of astigmatism.

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