非球面子孔径拼接干涉测量的几何方法研究
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摘要
非球面光学零件具有矫正像差、改善像质、扩大视场和增大作用距离的优点,同时还能够减轻系统重量,减小占用空间,因此在现代光学系统中得到了越来越广泛的应用。随着装备性能要求的不断增长,高能激光武器、激光核聚变和空间望远镜等武器装备光学系统对大型光学零件的需求激增,其技术要求上较传统光学零件也有很大提升,突出体现在对大口径、大相对口径光学零件的中高频误差提出了严格要求。全口径、全波段(有效口径内波前的各种空间频率成分)面形误差的检测成为大型光学零件检测的主要目标。这个问题目前并没有得到有效解决,子孔径测试技术是最有希望的解决方案。本论文研究工作的主要任务就是要有效解决非球面子孔径拼接测量的关键问题。与传统的子孔径拼接算法不同,论文从几何学观点出发,结合工件定位、公差评定以及图像多视拼合理论中的一些方法,充分利用计算机软件处理技术,对这些关键问题展开研究。论文的研究工作包括以下几个部分:
1.系统地研究了对称特征和多特征的位形空间理论,为形位公差和子孔径拼接问题提供了一个比较严格的数学背景。在此基础上按照美国标准ASME Y14.5.1M-1994的形式给出了形位公差的数学模型和评定算法,并且提出了改进算法性能的几个措施。与其他常用算法的性能比较和数值仿真验证了算法的有效性和优越性。
2.从几何学观点出发,对子孔径拼接问题进行数学建模,将问题分解为重叠计算子问题和测试几何参数优化子问题,它实际上可以看作是广义的工件定位或图像多视拼合问题。提出基于交替优化和序列线性化的迭代优化算法,结合处理海量数据的稀疏矩阵技术、顺序QR分解方法以及代码向量化等编程技巧,可以快速有效地求解非球面子孔径拼接问题。由于算法利用了被测曲面的设计模型简化重叠计算子问题,随之全口径也要相对于名义表面最佳定位,因此算法是子孔径拼接与工件定位的结合。算法的主要优点是对相当大的测试几何参数(包括位形参数)的不确定性不敏感,因此就不再需要精确已知调零与对准运动量,以及参考球面半径等几何参数。通过大口径抛物面镜的子孔径拼接仿真,验证了算法的有效性。进一步提出利用标记点作为辅助手段,进行对称自由度的位形参数优化,从而实现全部6个自由度的运动不确定性补偿。采用平面干涉仪进行测量的子孔径拼接问题,由于测量数据以参考平面为基准,因此比采用球面干涉仪的子孔径拼接问题更简单,或者可以看作是后者的一个特例。
3.子孔径划分是指确定子孔径的布局。由于曲率连续变化,非球面特别是离轴非球面的子孔径划分是一件精细而又复杂的工作。以凹抛物面为例,讨论了确定子孔径布局和计算子孔径最佳拟合球的方法,通过最小化用曲面积分表示的均方非球面偏差,确定最佳拟合球。随后给出了数值实例对提出的方法进行阐明和有效性验证。根据自由度分析确定了子孔径拼接干涉仪原型样机的设计方案,根据子孔径拼接算法对运动不确定性的鲁棒性,以及子孔径干涉图对运动误差的灵敏性分析,确定了对准与调零运动的分辨率、精度和行程等指标。
4.分析了子孔径拼接测量的主要误差因素,结合子孔径拼接算法原理,讨论了测量不确定度的传递关系。在一定的假设下,得到了单点相位测量误差不确定度到单点绝对相位法向误差的不确定度传递的解析式。
5.以平面镜和抛物面镜为例,划分子孔径后进行子孔径拼接干涉测量实验,并应用子孔径拼接算法获得全口径面形,与全口径测试得到的面形进行比较,对子孔径拼接测量的有效性进行实验验证。
Aspheric optics are being used more and more widely in modern optical systems,due to their ability of correcting aberrations, enhancing the image quality, enlarging thefield of view and extending the range of e?ect, while reducing the weight and volume ofthe system. With the ever-increasing demands on system performances, large optics areadopted in high-energy laser weapons, laser fusion systems and space telescopes. Thetechnical requirements are more than traditional. More rigorous control of high-middlefrequency error is now demanded within the full aperture of the large-relative apertureoptics. Thereby surface quality within the full aperture and full band of frequency (allspatial frequency components of the wavefront in the e?ective aperture) becomes themain content of inspection of large optics. Whereas this problem has never been solved.The subaperture testing method seems to provide the answer. This thesis is dedicatedto solve the key problems in subaperture testing of aspheric surfaces. Unlike traditionalsubaperture stitching algorithms, the problems are formulated from the geometrical pointof view, combined with methods for workpiece localization, tolerance assessment and multi-view registration. The major research e?orts include the following points.
1. A systematic introduction is first given to the theory of configuration space of sym-metric features and multi-features, which provides a mathematical background for theproblem of tolerance assessment and subaperture stitching. Then the mathematicalmodels and algorithms for tolerance assessment problem are proposed, in the form ofthe American standard ASME Y14.5.1M-1994. Several techniques are discussed toimprove the algorithm performances. Comparisons with other published algorithmsprove the validity and advantages of the proposed method.
2. A geometrical approach is introduced to formulate the subaperture stitching problemmathematically. The problem is decomposed into a series of overlapping calculationsubproblems and geometrical parameter optimization subproblems. Actually it canbe viewed as generalized workpiece localization or multi-view registration problem.By virtue of the alternating optimization technique and the successive linearizationmethod, the problem is solved e?ciently, combined with sparse techniques, sequentialQR decomposition method and code vectorization skills. The model of the surface un-der test is utilized to simplify the overlapping calculation subproblem. Consequentlythe full aperture should be best localized with regard to the nominal surface. There-fore the algorithm is a combination of subaperture stitching algorithm and workpiecelocalization algorithm. As a major advantage, the algorithm is immune from fairlybig parameter (including the motion parameter) uncertainties. Thanks to it, preciseprior knowledge of the nulling and alignment motion is no longer required, nor arethe radii of best fit spheres. Simulations of subaperture stitching test of a large-scaleparaboloid surface verify the validity of the proposed algorithms. Stitching algorithm with the aid of fiducial marks is also developed to further compensate uncertaintyof the symmetric degrees of freedom. Subaperture stitching problem using a pla-nar interferometer is simpler than using a spherical interferometer, since the phase ismeasured with a plane datum. It can be considered as special case of the latter.
3. Lattice is used here to mean the collection and arrangement of subapertures. Becauseof the varying curvature, lattice design is subtle and complicate for aspheric surfaces,especially for o?-axis subapertures. Methods are described for lattice design and cal-culation of the best fit sphere for each subaperture. The best fit sphere is determinedby minimizing the mean-squares aspheric deviations in the form of surface integral. Anumerical example is given to illustrate the procedure, and also verify the validity ofthe proposed methods. A prototype design of the subaperture stitching interferome-ter is presented based on the analysis of the degree-of-freedom. Requirements on theresolution, accuracy and stroke of the nulling and alignment motion are determinedby the robustness of the stitching algorithm against motion uncertainties, as well asthe sensitivity of sub-interferograms to motion errors.
4. Major error sources in subaperture stitching interferometry are recognized. The prop-agation of uncertainty is discussed with the subaperture stitching algorithms. Undercertain assumptions, it can be explicitly formulated.
5. Finally lattice design and subaperture stitching test are performed with a planar mir-ror and a paraboloid mirror respectively. The stitched full-aperture is then comparedwith the measured full-aperture, which verifies the validity of the subaperture testingexperimentally.
1.系统地研究了对称特征和多特征的位形空间理论,为形位公差和子孔径拼接问题提供了一个比较严格的数学背景。在此基础上按照美国标准ASME Y14.5.1M-1994的形式给出了形位公差的数学模型和评定算法,并且提出了改进算法性能的几个措施。与其他常用算法的性能比较和数值仿真验证了算法的有效性和优越性。
2.从几何学观点出发,对子孔径拼接问题进行数学建模,将问题分解为重叠计算子问题和测试几何参数优化子问题,它实际上可以看作是广义的工件定位或图像多视拼合问题。提出基于交替优化和序列线性化的迭代优化算法,结合处理海量数据的稀疏矩阵技术、顺序QR分解方法以及代码向量化等编程技巧,可以快速有效地求解非球面子孔径拼接问题。由于算法利用了被测曲面的设计模型简化重叠计算子问题,随之全口径也要相对于名义表面最佳定位,因此算法是子孔径拼接与工件定位的结合。算法的主要优点是对相当大的测试几何参数(包括位形参数)的不确定性不敏感,因此就不再需要精确已知调零与对准运动量,以及参考球面半径等几何参数。通过大口径抛物面镜的子孔径拼接仿真,验证了算法的有效性。进一步提出利用标记点作为辅助手段,进行对称自由度的位形参数优化,从而实现全部6个自由度的运动不确定性补偿。采用平面干涉仪进行测量的子孔径拼接问题,由于测量数据以参考平面为基准,因此比采用球面干涉仪的子孔径拼接问题更简单,或者可以看作是后者的一个特例。
3.子孔径划分是指确定子孔径的布局。由于曲率连续变化,非球面特别是离轴非球面的子孔径划分是一件精细而又复杂的工作。以凹抛物面为例,讨论了确定子孔径布局和计算子孔径最佳拟合球的方法,通过最小化用曲面积分表示的均方非球面偏差,确定最佳拟合球。随后给出了数值实例对提出的方法进行阐明和有效性验证。根据自由度分析确定了子孔径拼接干涉仪原型样机的设计方案,根据子孔径拼接算法对运动不确定性的鲁棒性,以及子孔径干涉图对运动误差的灵敏性分析,确定了对准与调零运动的分辨率、精度和行程等指标。
4.分析了子孔径拼接测量的主要误差因素,结合子孔径拼接算法原理,讨论了测量不确定度的传递关系。在一定的假设下,得到了单点相位测量误差不确定度到单点绝对相位法向误差的不确定度传递的解析式。
5.以平面镜和抛物面镜为例,划分子孔径后进行子孔径拼接干涉测量实验,并应用子孔径拼接算法获得全口径面形,与全口径测试得到的面形进行比较,对子孔径拼接测量的有效性进行实验验证。
Aspheric optics are being used more and more widely in modern optical systems,due to their ability of correcting aberrations, enhancing the image quality, enlarging thefield of view and extending the range of e?ect, while reducing the weight and volume ofthe system. With the ever-increasing demands on system performances, large optics areadopted in high-energy laser weapons, laser fusion systems and space telescopes. Thetechnical requirements are more than traditional. More rigorous control of high-middlefrequency error is now demanded within the full aperture of the large-relative apertureoptics. Thereby surface quality within the full aperture and full band of frequency (allspatial frequency components of the wavefront in the e?ective aperture) becomes themain content of inspection of large optics. Whereas this problem has never been solved.The subaperture testing method seems to provide the answer. This thesis is dedicatedto solve the key problems in subaperture testing of aspheric surfaces. Unlike traditionalsubaperture stitching algorithms, the problems are formulated from the geometrical pointof view, combined with methods for workpiece localization, tolerance assessment and multi-view registration. The major research e?orts include the following points.
1. A systematic introduction is first given to the theory of configuration space of sym-metric features and multi-features, which provides a mathematical background for theproblem of tolerance assessment and subaperture stitching. Then the mathematicalmodels and algorithms for tolerance assessment problem are proposed, in the form ofthe American standard ASME Y14.5.1M-1994. Several techniques are discussed toimprove the algorithm performances. Comparisons with other published algorithmsprove the validity and advantages of the proposed method.
2. A geometrical approach is introduced to formulate the subaperture stitching problemmathematically. The problem is decomposed into a series of overlapping calculationsubproblems and geometrical parameter optimization subproblems. Actually it canbe viewed as generalized workpiece localization or multi-view registration problem.By virtue of the alternating optimization technique and the successive linearizationmethod, the problem is solved e?ciently, combined with sparse techniques, sequentialQR decomposition method and code vectorization skills. The model of the surface un-der test is utilized to simplify the overlapping calculation subproblem. Consequentlythe full aperture should be best localized with regard to the nominal surface. There-fore the algorithm is a combination of subaperture stitching algorithm and workpiecelocalization algorithm. As a major advantage, the algorithm is immune from fairlybig parameter (including the motion parameter) uncertainties. Thanks to it, preciseprior knowledge of the nulling and alignment motion is no longer required, nor arethe radii of best fit spheres. Simulations of subaperture stitching test of a large-scaleparaboloid surface verify the validity of the proposed algorithms. Stitching algorithm with the aid of fiducial marks is also developed to further compensate uncertaintyof the symmetric degrees of freedom. Subaperture stitching problem using a pla-nar interferometer is simpler than using a spherical interferometer, since the phase ismeasured with a plane datum. It can be considered as special case of the latter.
3. Lattice is used here to mean the collection and arrangement of subapertures. Becauseof the varying curvature, lattice design is subtle and complicate for aspheric surfaces,especially for o?-axis subapertures. Methods are described for lattice design and cal-culation of the best fit sphere for each subaperture. The best fit sphere is determinedby minimizing the mean-squares aspheric deviations in the form of surface integral. Anumerical example is given to illustrate the procedure, and also verify the validity ofthe proposed methods. A prototype design of the subaperture stitching interferome-ter is presented based on the analysis of the degree-of-freedom. Requirements on theresolution, accuracy and stroke of the nulling and alignment motion are determinedby the robustness of the stitching algorithm against motion uncertainties, as well asthe sensitivity of sub-interferograms to motion errors.
4. Major error sources in subaperture stitching interferometry are recognized. The prop-agation of uncertainty is discussed with the subaperture stitching algorithms. Undercertain assumptions, it can be explicitly formulated.
5. Finally lattice design and subaperture stitching test are performed with a planar mir-ror and a paraboloid mirror respectively. The stitched full-aperture is then comparedwith the measured full-aperture, which verifies the validity of the subaperture testingexperimentally.
引文
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