Simulation and measurement of systematic errors of stitching interferometry for high precision X-ray mirrors with large radius of curvature

To accurately measure the surface figure of curved mirrors with large radius of curvature (RoC) using stitching interferometry methods, three types of measurement errors are systematically studied, including retrace error, defocusing error within a single subaperture, and stitching angle error among different subapertures. It was found that part of the retrace error caused by the mismatch between the reference wavefront and reflected wavefront has little effect, while the overall retrace error, including the influence of the imperfect optical elements, will cause an error of 1–2 nm RMS within a single subaperture. Defocusing error will enlarge the error due to amplification of optical path error caused by the deviation of the position of the CCD. Because the error is mainly in the edge area, a slope threshold, which controls the maximum surface slope of each subaperture, can be optimized to reduce the effect of the defocus on stitching measurement error. Constant angle error among neighboring subapertures has the biggest accumulation effect on the final stitched figure. For the spherical mirror with RoC of 100 m of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>80</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>40</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> , the error of the one-dimensional residual profile is 4.67 nm PV, assuming a constant angle error of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>7</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">a</mml:mi> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> </mml:math> . For the elliptical mirror with RoC of 60–140 m, it is more than 15 nm peak-to-valley (PV). It is because the profile difference caused by constant angle error is closer to a circle, which can be mostly removed after subtraction of a best-fit sphere. Based on the above error analysis, the developed algorithm-based stitching method was used to measure an elliptical cylindrical mirror of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>74</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>40</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> with RoC of 60–140 m, and the result was compared with a slope measurement instrument from the Beijing Synchrotron Radiation Facility. After removing the best fitting ellipse profile, the one-dimensional difference between the two results is only 0.77 nm RMS, which demonstrated good measurement accuracy.