﻿ 拼接检测系统平面波前稀疏子孔径排列模型的优化
 光电工程  2018, Vol. 45 Issue (5): 170638      DOI: 10.12086/oee.2018.170638

1. 中国科学院光电技术研究所，四川 成都 610209;
2. 中国科学院大学，北京 100049

Optimization of sparse subaperture array model for stitching detection of plane wavefront
Luo Qian1,2, Wu Shibin1, Wang Lihua1, Yang Wei1, Fan Bin1
1. Institute of Optics and Electronics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
2. University of Chinese Academy of Sciences, Beijing 100049, China
Abstract: The sparse subaperture stitching, the accuracy of which is closely related to the arrangement, number and size of subapertures, is one of the main methods of quality testing for large aperture optical systems. A mathematical model was established to deduce the relation curve between the subaperture number k and fill factor M when the value of k ranges from one to infinity. As a result, the optimal arrangement layout, consisting of seven sparse subapertures, was obtained for the detection systems below 1.5 m. Autocollimation interference detection of Φ200 mm validated the rationality of the arrangement.
Keywords: sparse subaperture    mathematical modeling    stitching detection    wavefront reconstruction    interferometry

1 引言

2 稀疏子孔径排列模型

 图 1 稀疏子孔径检测的工作原理 Fig. 1 Schematic diagram of sparse subaperture detection

1) 同样数目、同样大小的子孔径排列方式不同，其获取的波前信息也有差别，子孔径对称分布优于非对称分布，子孔径均匀分布优于非均匀分布。

2) 同样数目、同样排列方式的稀疏子孔径，子孔径大小就成了影响检测精度的关键因素。

 $M = \frac{{{S_k}}}{{{S_0}}} = \frac{{k{\rm{ \mathit{ π} }}{r^2}}}{{{\rm{ \mathit{ π} }}{R^2}}},$ (1)

1) 首先在最外层放一圈子孔径，均与全口径内切，两两子孔径之间相互外切。

2) 将全口径的半径减去子孔径的直径，所得的值大于子孔径直径，则再放置一层子孔径，相互外切，并与上一层的子孔径外切。

3) 将全口径的半径减去2倍子孔径的直径，所得的值大于子孔径直径，则再放置一层子孔径，相互外切，并与上一层的子孔径外切。

4) 以此类推，直到剩余部分只能放下一个子孔径为止。

 图 2 蜂窝状的子孔径排列图 Fig. 2 Honeycomb-shaped arrangement of subapertures

1) 当$\frac{1}{2} < r \le 1$时，k=1，此时，只能放置一个子孔径，且R=rM=1；

2) 当$\frac{1}{3} < r \le \frac{1}{2}$时，k取2~7，如表 1所示，此时，子孔径只能沿着全口径围成一圈；

 k 2 3 4 5 6 7 M 0.5 0.64617 0.68629 0.68521 0.66667 0.77778

3) 当$\frac{1}{5} < r \le \frac{1}{3}$时，k取8~19，如表 2所示，此时，子孔径沿着全口径围成一圈，中间部分不能放下第二层子孔径；

 k 8 9 10 11 12 13 14 15 16 17 18 19 M 0.732 0.689 0.649 0.613 0.668 0.724 0.676 0.724 0.676 0.718 0.671 0.76

a) 当$\frac{1}{4} < r \le \frac{1}{3}$时，k取8~10，中间部分仅能放置一个子孔径；

b) 当$\frac{1}{5} < r \le \frac{1}{4}$时，k取11~19，中间部分能放置不止一个子孔径。

 图 3 子孔径个数与填充因子的关系曲线图(k取1~19) Fig. 3 The relation curve between subaperture number and filling factor (k=1~19)

 图 4 子孔径个数k与填充因子M的关系曲线(k取1~∞) Fig. 4 The relation curve between subaperture number and filling factor (k=1~∞)
 $M = \mathop {{\rm{lim}}}\limits_{n \to \infty } [1 + 3n(n + 1)] \times {\left( {\frac{2}{{2 + 4n}}} \right)^2} = 0.75。$

n取1到无穷。

 图 5 最优排列布局 Fig. 5 Optimal arrangement
3 四个子孔径到九个子孔径的实验结果对比

 图 6 实验装置图 Fig. 6 Photograph of experimental setup

 图 7 四到九个子孔径的稀疏子孔径排列方式 Fig. 7 Sparse subaperture arrangement diagrams with four to nine subapertures

 图 8 去除相位常数、倾斜后的面形图 Fig. 8 Surface pattern after removing phase constant andtilt

 图 9 四到九个子孔径拼接检测的全孔径波前图。(a)四个子孔径；(b)五个子孔径；(c)六个子孔径；(d)七个子孔径；(e)八个子孔径；(f)九个子孔径 Fig. 9 Full aperture wavefront mapping for multiple subaperture stitching detection. (a) Four subapertures; (b) Five subapertures; (c) Six subapertures; (d) Seven subapertures; (e) Eight subapertures; (f) Nine subapertures

 图 10 四到九个子孔径拼接检测与直接检测波前的残差图。(a)四个子孔径；(b)五个子孔径；(c)六个子孔径；(d)七个子孔径；(e)八个子孔径；(f)九个子孔径 Fig. 10 Wavefront residuals between stitching and direct detection with multiple subapertures. (a) Four subapertures; (b) Five subapertures; (c) Six subapertures; (d) Seven subapertures; (e) Eight subapertures; (f) Nine subapertures

 No. Wavefront PV/λ Wavefront RMS/λ Wavefront residuals PV/λ Wavefront residuals RMS/λ 4 0.2369 0.0377 0.2150 0.0169 5 0.1781 0.0337 0.0987 0.0107 6 0.2136 0.0433 0.1039 0.0103 7 0.1901 0.0391 0.1160 0.0092 8 0.2147 0.0399 0.1288 0.0109 9 0.2069 0.0363 0.1186 0.0110

4 结论

 [1] Hou X, Wu F, Yang L, et al. Status and development trend of sub-aperture stitching interferometric testing technique[J]. Optics & Optoelectronic Technology, 2005, 3(3): 50-53. 侯溪, 伍凡, 杨力, 等. 子孔径拼接干涉测试技术现状及发展趋势[J]. 光学与光电技术, 2005, 3(3): 50-53. [2] Sj dahl M, Oreb B F. Stitching interferometric measurement data for inspection of large optical components[J]. Optical Engineering, 2002, 41(2): 403-408. DOI:10.1117/1.1430727 [3] Kim C J. Polynomial fit of interferograms[J]. Applied Optics, 1982, 21(24): 4521-4525. DOI:10.1364/AO.21.004521 [4] He Y, Wang Z X, Wang Q, et al. Testing the large aperture optical components by the sub-aperture stitching interferometer[J]. Proceedings of SPIE, 2007, 6624: 66240D. [5] Xu X D, Shen Z X, Tong G D, et al. Sparse subaperture stitching method for measuring large aperture planar optics[J]. Optical Engineering, 2016, 55(2): 024103. DOI:10.1117/1.OE.55.2.024103 [6] Yan F T, Fan B, Hou X, et al. Large-aperture mirror test using sparse sub-aperture sampling[J]. High Power Laser and Particle Beams, 2011, 23(12): 3193-3196. 闫锋涛, 范斌, 侯溪, 等. 稀疏子孔径采样检测大口径光学器件[J]. 强激光与粒子束, 2011, 23(12): 3193-3196. [7] Chow W W, Lawrence G N. Method for subaperture testing interferogram reduction[J]. Optics Letters, 1983, 8(9): 468-470. DOI:10.1364/OL.8.000468 [8] Wang Y, Fu R M, Liao Z B. Wavefront reconstruction algorithm based on sparse aperture[J]. Spacecraft Recovery & Remote Sensing, 2015, 36(5): 51-59. 王琰, 伏瑞敏, 廖志波. 基于稀疏孔径的波前重构算法[J]. 航天返回与遥感, 2015, 36(5): 51-59. [9] Chen S Y, Dai Y F, Li S Y, et al. Error reductions for stitching test of large optical flats[J]. Optics & Laser Technology, 2012, 44(5): 1543-1550. [10] Smith G A, Burge J H. Subaperture stitching surface errors due to noise[J]. Proceedings of SPIE, 2015, 9575: 95750W. DOI:10.1117/12.2188085 [11] Hill J M, Salinari P. The large binocular telescope project[J]. Proceedings of SPIE, 2000, 5489: 603-614. [12] Chung S J, Miller D W, De Weck O L. Design and implementation of sparse aperture imaging systems[J]. Proceeding of SPIE, 2002, 4849: 181-192. DOI:10.1117/12.460077