Cylindrical optics stitching interferometry with low spatial frequency figure error correction
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摘要:
使用基于计算全息图（CGH）的子孔径拼接检测方法是检测柱面镜表面轮廓的常用方法之一。由于误差的累积放大，以及基于正交多项式的常规像差拟合方法无法有效分离误差和真实面形，拼接结果存在低频面形信息失真的问题。针对这一现象，本文提出了一种补偿修正柱面镜表面轮廓低频面形信息的新方法，先使用基于逐次拼接的切比雪夫多项式拼接方法进行拼接，再单独对镜面的拼接方向（即母线方向）轮廓进行检测，获取母线方向低频信息后，对拼接结果进行进一步的融合修正。通过对一通光口径为150 mm×210 mm，曲率半径为790.23 mm的柱面反射镜进行拼接检测实验验证，所提方法有效修正了柱面镜的母线方向轮廓，与使用全口径CGH 检测得到的全口径参考面形相比，柱面镜拼接检测结果残差均方根（RMS值）约为
0.0103 λ，与修正前相比，拼接残差RMS值降低了约37%，检测精度显著提升。方法具备操作简便、硬件要求较低、检测精度可靠等优势。Abstract:The subaperture stitching method based on computer-generated holograms (CGH) is a common approach for measuring the surface profile of cylindrical mirrors. However, the stitching result suffers from distortion in low-frequency surface shape information. This is primarily caused by the cumulative amplification of errors and the inability of conventional aberration fitting methods (based on orthogonal polynomials) to effectively separate errors from the true surface figure. To address this issue, this paper proposes a novel method to compensate for and correct the low-frequency information of cylindrical mirror surface profiles. First, an initial stitching is performed using a successive subaperture stitching method based on Chebyshev polynomials. Next, the profile along the mirror's stitching direction (i.e., the generatrix direction) is measured independently to extract its low-frequency component. Finally, this low-frequency information is used to further fuse and correct the initial stitching result. Experimental validation was conducted on a cylindrical mirror with a clear aperture of 150 mm × 210 mm and a radius of curvature of 790.23 mm. The results demonstrate that the proposed method effectively corrects the generatrix direction profile of the cylindrical mirror. Compared to the full-aperture reference surface obtained via full-aperture CGH measurement, the root mean square (RMS) of the residual error for the stitching result is approximately
0.0103 λ. This represents a reduction of about 37% in the RMS value compared to the pre-correction result, indicating a significant improvement in measurement accuracy. The proposed method offers advantages including ease of implementation, low hardware requirements, and reliable measurement accuracy. -
图 2 柱面镜局部面形与低频像差耦合示意图：(a)仿真的全口径母线方向轮廓（已去除倾斜和离焦项）；(b)截取的局部面形轮廓；(c)局部面形中类似倾斜与离焦的分量
Figure 2. Schematic diagram of coupling between the local surface shape of the cylindrical mirror and low-frequency aberration: (a) simulated full-aperture generatrix direction profile; (b) Extracted local surface shape profile; (c) the components similar to tilt and defocus in the local surface
图 3 不同方向类离焦误差对拼接精度的影响对比（注：数据点为十次仿真结果的平均残差RMS与引入误差PV值的关系，拼接结果已去除初阶像差）
Figure 3. Comparison of stitching accuracy affected by defocus-like errors in stitching and curvature directions. (Note: Each data point represents the average residual RMS of ten simulations versus the introduced PV value; low-order aberrations have been removed from the stitching results.)
图 4 低频面形融合修正方法原理示意图
Figure 4. Schematic diagram of the low-frequency surface shape fusion correction method
图 7 实验检测结果：(a)子孔径检测结果(自上而下：#1、#2、#3)；(b)柱面镜拼接方向轮廓检测结果
Figure 7. Experimental test results: (a) Sub-aperture measurement results (from top to bottom: #1, #2, #3); (b) Measured profile of the cylindrical mirror along the stitching direction
图 8 低频面形修正效果(注：中部区域母线方向轮廓数据通过对柱面镜中部100列数据点取平均获得)
Figure 8. Correction effect of low-frequency surface shape. (Note: The profile along the generatrix direction at the central region is obtained by averaging 100 data columns across the cylinder center.)
图 9 拼接检测结果与CGH检测结果对比
Figure 9. Comparison of stitching measurement and CGH measurement results
表 1 切比雪夫多项式各阶表达式与像差的对应关系
Table 1. The correspondence between Chebyshev polynomials and aberrations
阶数 多项式 x轴 多项式 y轴 0 1 平移 1 平移 1 x 倾斜 y 倾斜 2 2x2-1 离焦 2y2-1 离焦 3 4x3-3x 慧差 4y3-3y 慧差 4 8x4-8x2+1 球差 8y4-8y2+1 球差 
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表 2 柱面子孔径拼接检测实验数据
Table 2. Experimental data for cylindrical subaperture stitching
残差/λ 低频面形修正前 低频面形修正后 RMS 0.0163 0.0103 PV 0.1375 0.1025
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表 3 检测精度分析
Table 3. Measurement Accuracy Analysis (Unit: nm)
误差类型 误差大小 随机误差 2.5 干涉仪误差 1.0 CGH误差 3.0 拼接算法精度 6.7 检测精度 7.8
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