自由曲面表征函数及其应用研究
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摘要
自由曲面是指非旋转对称的、各处曲率不同的光学表面,针对自由曲面的研究是近年来光学界的热点之一。研究方向包括自由曲面表征函数、自由曲面光学系统设计、自由曲面加工和检测,自由曲面像差理论等方面。目前自由曲面已经被逐渐地应用在光学工程领域,但随着设计视场、相对孔径增大,自由曲面光学系统非对称高阶像差在逐步增大。然而当前的自由曲面表征函数,如泽尼克、高斯径向基函数和Q-TYPE函数等不能提供更多空间变化自由度来平衡逐渐增大的非对称像差,所以限制了自由曲面在高分辨率反射望远系统中的应用。因此,构建新型自由曲面表征函数对于自由曲面的广泛应用具有重要意义。
对于离轴自由曲面光学系统设计设计者来说,构建合适的初始结构需要消耗大量的时间。因此,提出一种快速的离轴自由曲面光学系统初始结构设计方法对于光学设计也同样具有重要意义。
近些年,在具体工程中采用方形和六边形的非圆形孔径的离轴自由曲面光学系统的应用越来越广泛。在非圆形孔径下泽尼克多项式并不正交,在离轴自由曲面光学系统装调中采用何种函数解析非圆形孔径离轴自由曲面光学系统的波前；所有以上问题均为自由曲面表征函数应用于光学系统时所必须要解决的问题。
本论文基于以上问题针对自由曲面表征函数及其应用进行研究,目的在于解决目前表征函数对于大多数面型表征和优化能力的不足的问题；解决离轴自由曲面光学系统初始结构构建耗时太长的问题；解决在非圆形孔径下自由曲面系统装调时泽尼克多项式不正交的问题。主要的研究内容总结为以下几个方面：
(1)采用理论分析方法,对当前主流自由曲面表征函数,如泽尼克多项式、Q-TYPE、切比雪夫多项式和高斯径向基函数等的构造理论、优点和缺点,以及与像差项的对应关系进行深入研究；在此基础上构建新型表征函数-新函数；并提出了非圆形域正交多项式构造理论。而且还从理论上解释了节点像差理论不适于指导带有偏心和倾斜量面型的离轴自由曲面光学系统装调。
(2)采用数值分析方法研究第二章中所提到的自由曲面表征函数,分析自由曲面表征函数对于选取的具有代表性的自由曲面拟合精度和优化能力,以突出新型表征函数-新函数对于各种面型拟合精度的绝对优势；并利用数值分析方法证明不能直接采用泽尼克多项式解析系统波前。
(3)针对离轴自由曲面光学系统初始结构构建耗时长的问题,提出了一种快速的离轴自由曲面初始结构构造理论；结合新函数设计了一种大视场(60。)离轴自由曲面光学系统,对此光学系统成像质量和采用面型的可加工性进行了分析,分析结果满足指标要求。
(4)利用方形域正交多项式-勒让德正交多项式解析方形孔径系统波前,结合灵敏度矩阵方法对设计的方形孔径超大视场离轴自由曲面系统进行装调,装调结果表明了此装调方法的正确性,对于其他非圆形离轴自由曲面光学系统装调具有借鉴意义。
Free-form surface, which is non-rotational symmetric optical surface, has become a hot research area in the optical industry. The direction of research contains the free-form surface characterization function, the free-form surface optical system design, the free-form surface fabrication with testing, and the free-form surface aberration theory. At present, free-form surface is widely and gradually used in the field of optical engineering. However, with the enhancing requirement of large designing field of view (FOV), relative aperture enlargement, and imaging quality, the non-rotational high-order aberration of the free-form surface system is increased step by step. But the current characterization functions of free-form surface, such as Zernike Function, Gaussian Radial Basis Function (GRBF), and the Q-TYPE Function, cannot provide enough spatial variables to balance the gradually increased non-rotational aberration, which has limited the application of the free-form surface in the system of high-resolution reflecting telescope. Therefore, the creation of new-type characterization function has big significance on the widespread use of free-form surface.
It takes a long time to construct the appropriate initial structure for the designers of the off-axis free-form surface system. Therefore, it is also greatly significant to propose an efficient approach to build initial structure of the off-axis free-form surface system for optical design.
In recent years, the rectangle and hexagon surfaces, which are non-circular aperture, are widely used in the off-axis free-form surface system. The Zernike Polynomial is not orthogonal over the non-circular aperture, and which function should be utilized to analyze wave-front of the off-axis free-form surface system with non-circular aperture in alignment. All the problems mentioned above must be settled when the free-form surface is applied in the off-axis free-form surface system. This dissertation refers to the research on the free-form surface characterization function and the application of the free-form surface characterization function based on the problems mentioned above. The destination is listed as follows:solve the limitation of surface characterization ability and optimization ability by using most of the existing functions; solve the time-consuming problem when build the initial system structure; solve the non-orthogonal features of the Zernike Polynomial which is used to align off-axis free-from surface system with non-circular aperture. The research contents are mainly explained as follows:
(1) By using theoretical analyzing method, firstly, the theory, the advantages and disadvantages, and the corresponding relationship with aberration items of the popularly characterization functions of free-form surface, such as Zernike Polynomial, Q-TYPE, Chebyshev Polynomial, and GRBF are researched deeply. Secondly, one new-type characterization function named New function is established on the basis of the research above. In the meantime, the non-circular orthogonal polynomial theory is proposed. At last, the reason why nodal aberration is not appropriate to guide the alignment of the free-form surface with decentration and tilt is explained.
(2) The free-form surface characterization function mentioned in the Section Two is researched by using numerical analysis method. For the selected typical free-form surface, the fitting precision and the optimization ability by using the characterization function are analyzed in order to highlight the fitting precision advantage of the New function. And it is proved that the Zernike Polynomial cannot be directly used in resolving by means of numerical analysis.
(3) Focusing on the time consuming problem of constructing initial free-form surface optical system structure, an efficient method for structuring the off-axis free-form surface is proposed. Then, one large field of view (60°) off-axis free-from surface system is designed by integrating New function. The imaging quality and the processing possibility of the optical system are analyzed. The result meets the requirement.
(4) The rectangle aperture wave-front is analyzed by using rectangle area orthogonal polynomial named Legendre Polynomial. By integrating the sensitivity matrix method, the rectangle aperture free-form surface system with large field of view is aligned. The result shows that the alignment method is correct, which can use for reference to other non-circular aperture off-axis free-form surface system alignment.
对于离轴自由曲面光学系统设计设计者来说,构建合适的初始结构需要消耗大量的时间。因此,提出一种快速的离轴自由曲面光学系统初始结构设计方法对于光学设计也同样具有重要意义。
近些年,在具体工程中采用方形和六边形的非圆形孔径的离轴自由曲面光学系统的应用越来越广泛。在非圆形孔径下泽尼克多项式并不正交,在离轴自由曲面光学系统装调中采用何种函数解析非圆形孔径离轴自由曲面光学系统的波前；所有以上问题均为自由曲面表征函数应用于光学系统时所必须要解决的问题。
本论文基于以上问题针对自由曲面表征函数及其应用进行研究,目的在于解决目前表征函数对于大多数面型表征和优化能力的不足的问题；解决离轴自由曲面光学系统初始结构构建耗时太长的问题；解决在非圆形孔径下自由曲面系统装调时泽尼克多项式不正交的问题。主要的研究内容总结为以下几个方面：
(1)采用理论分析方法,对当前主流自由曲面表征函数,如泽尼克多项式、Q-TYPE、切比雪夫多项式和高斯径向基函数等的构造理论、优点和缺点,以及与像差项的对应关系进行深入研究；在此基础上构建新型表征函数-新函数；并提出了非圆形域正交多项式构造理论。而且还从理论上解释了节点像差理论不适于指导带有偏心和倾斜量面型的离轴自由曲面光学系统装调。
(2)采用数值分析方法研究第二章中所提到的自由曲面表征函数,分析自由曲面表征函数对于选取的具有代表性的自由曲面拟合精度和优化能力,以突出新型表征函数-新函数对于各种面型拟合精度的绝对优势；并利用数值分析方法证明不能直接采用泽尼克多项式解析系统波前。
(3)针对离轴自由曲面光学系统初始结构构建耗时长的问题,提出了一种快速的离轴自由曲面初始结构构造理论；结合新函数设计了一种大视场(60。)离轴自由曲面光学系统,对此光学系统成像质量和采用面型的可加工性进行了分析,分析结果满足指标要求。
(4)利用方形域正交多项式-勒让德正交多项式解析方形孔径系统波前,结合灵敏度矩阵方法对设计的方形孔径超大视场离轴自由曲面系统进行装调,装调结果表明了此装调方法的正确性,对于其他非圆形离轴自由曲面光学系统装调具有借鉴意义。
Free-form surface, which is non-rotational symmetric optical surface, has become a hot research area in the optical industry. The direction of research contains the free-form surface characterization function, the free-form surface optical system design, the free-form surface fabrication with testing, and the free-form surface aberration theory. At present, free-form surface is widely and gradually used in the field of optical engineering. However, with the enhancing requirement of large designing field of view (FOV), relative aperture enlargement, and imaging quality, the non-rotational high-order aberration of the free-form surface system is increased step by step. But the current characterization functions of free-form surface, such as Zernike Function, Gaussian Radial Basis Function (GRBF), and the Q-TYPE Function, cannot provide enough spatial variables to balance the gradually increased non-rotational aberration, which has limited the application of the free-form surface in the system of high-resolution reflecting telescope. Therefore, the creation of new-type characterization function has big significance on the widespread use of free-form surface.
It takes a long time to construct the appropriate initial structure for the designers of the off-axis free-form surface system. Therefore, it is also greatly significant to propose an efficient approach to build initial structure of the off-axis free-form surface system for optical design.
In recent years, the rectangle and hexagon surfaces, which are non-circular aperture, are widely used in the off-axis free-form surface system. The Zernike Polynomial is not orthogonal over the non-circular aperture, and which function should be utilized to analyze wave-front of the off-axis free-form surface system with non-circular aperture in alignment. All the problems mentioned above must be settled when the free-form surface is applied in the off-axis free-form surface system. This dissertation refers to the research on the free-form surface characterization function and the application of the free-form surface characterization function based on the problems mentioned above. The destination is listed as follows:solve the limitation of surface characterization ability and optimization ability by using most of the existing functions; solve the time-consuming problem when build the initial system structure; solve the non-orthogonal features of the Zernike Polynomial which is used to align off-axis free-from surface system with non-circular aperture. The research contents are mainly explained as follows:
(1) By using theoretical analyzing method, firstly, the theory, the advantages and disadvantages, and the corresponding relationship with aberration items of the popularly characterization functions of free-form surface, such as Zernike Polynomial, Q-TYPE, Chebyshev Polynomial, and GRBF are researched deeply. Secondly, one new-type characterization function named New function is established on the basis of the research above. In the meantime, the non-circular orthogonal polynomial theory is proposed. At last, the reason why nodal aberration is not appropriate to guide the alignment of the free-form surface with decentration and tilt is explained.
(2) The free-form surface characterization function mentioned in the Section Two is researched by using numerical analysis method. For the selected typical free-form surface, the fitting precision and the optimization ability by using the characterization function are analyzed in order to highlight the fitting precision advantage of the New function. And it is proved that the Zernike Polynomial cannot be directly used in resolving by means of numerical analysis.
(3) Focusing on the time consuming problem of constructing initial free-form surface optical system structure, an efficient method for structuring the off-axis free-form surface is proposed. Then, one large field of view (60°) off-axis free-from surface system is designed by integrating New function. The imaging quality and the processing possibility of the optical system are analyzed. The result meets the requirement.
(4) The rectangle aperture wave-front is analyzed by using rectangle area orthogonal polynomial named Legendre Polynomial. By integrating the sensitivity matrix method, the rectangle aperture free-form surface system with large field of view is aligned. The result shows that the alignment method is correct, which can use for reference to other non-circular aperture off-axis free-form surface system alignment.
引文
[1]Virendra N.Mahajan, Guang-ming Dai. Orthonormal polynomials in wavefront analysis:analytical solution[J]. J.Opt.Soc.Am.A,2007,24(9),2994-3016
[2]G.W.Forbes. Characterizing the shape of freeform optics[J]. OPTICS EXPRESS, 2012,20(3):2483-2449
[3]Ozan Cakmakci, Ilhan Kaya, Gregeory E.Fasshauer, Kevin P.Thompson, Jannick P.Rolland. Application of Radial Basis Functions to Represent Optical Freeform Surfacees[J]. Proc. of SPIE,2010,7652:76520A(1-8)
[4]张以谟.应用光学[M].北京：电子工业出版社,2008.494-494
[5]杨波,王涌天.自由曲面反射器的计算机辅助设计[J].光学学报,2004,24(6)：721-724
[6]陈云亮,李铁才,邱祥辉.头戴显示器中自由曲面棱镜的设计[J].应用光学,2009,30(4)：552-557
[7]Dewen C, Yongtian W, Hong H, et al. Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism[J]. APPLIED OPTICS,2009,48(14):2655-2668.
[8]Dewen C, Yongtian W, Hong H. Free form optical system design with differential equations[J]. Proc. of SPIE,2010,7849:78490Q(1-8)
[9]Ozan C, Gregory E F, Hassan F, et al. Meshfree Approximation Methods for Free-Form Surface Representation in Optical Design with Applications to Head-Worn Displays[J]. Proc. of SPIE,2008,7061:70610D(1-15)
[10]Ozan C, Brendan M, Hassan F, et al. Optimal local shape description for rotationally non-symmetric optical surface design and analysis[J]. OPTICS EXPRESS, 2008,16(3):1583-1589.
[11]孙旭涛.应用自由曲面的超薄投影显示系统理论和实验研究[D]：[博士学位论文].杭州：浙江大学,2008
[12]Lei L, Allen Y Y. Design and fabrication of a freeform prism array for 3D microscopy[J]. J. Opt. Soc. Am. A,2010,27(12):2613-2620.
[13]R A Hicks, Christopher C. Designing coupled free-form surfaces[J]. J. Opt. Soc. Am. A,2010,27(10):2132-2137
[14]Tao M, Jingchi Y, Pei L, et al. Design of a freeform varifocal panoramic optical system with specified annular center of field of view[J]. OPTICS EXPRESS,2011, 19(5):3843-3853
[15]J Buchheister, L Mueller. Reflex free double pass objective due to reflective free form surfaces[J]. Proc. of SPIE,2008,7100:71000N(1-10)
[16]R A Hicks. Direct Methods for Freeform Surface Design[J]. Proc. of SPIE,2007, 6668:666802(1-10)
[17]Peter O. Optic design of head-up displays with freeform surfaces specified by NURBS[J]. Proc. of SPIE,2008,7100:71000Y(1-12)
[18]梁敏勇,廖宁放,冯洁,等.三反射式柱面光学系统设计及优化[J].光学学报,2008,28(7)：1359-1363
[19]Kevin T. Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry[J]. J. Opt. Soc. Am. A,2005,22(7):1389-1401
[20]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:spherical aberration [J]. J. Opt. Soc. Am. A,2009,26(5): 1090-1100
[21]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:the astigmatic aberrations [J]. J.Opt.Soc.Am.A,2011,28(5):821-836
[22]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:the comatic aberrations[J]. J.Opt.Soc.Am.A,2010,27(6): 1490-1504
[23]Kevin P T, Tobias S, Ozan C. Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry[J]. J. Opt. Soc. Am. A,2009,26(6):1503-1517
[24]John R R. Techniques and tools for obtaining symmetrical performance from tilted-component systems[J]. Optical Engineering,2000,39(7):1776-1787
[25]王灵杰,张新,张建萍,史广维.自由曲面空间光学系统设计[J].2012,33(6)：1040-1046
[26]李庆扬,王能超,易大义.数值分析[M]。北京：清华大学出版社,2008.73-76
[27]王忠仁,张静.复变函数与积分变换[M].北京：高等教育出版社2006.113-116
[28]Tobias S, Kevin P T, Jannick P R. A unique astigmatic nodal property in misaligned Ritchey-Chretien telescopes with misalignment coma removed[J]. OPTICS EXPRESS,2010,18(5):5282-5288
[29]Tobias S, Kevin P T, Jannick P R. Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes[J]. APPLIED OPTICS,49(16):D131-D144
[30]Kevin P T, Tobias S, Jannick P R. The misalignment induced aberrations of TMA telescopes[J]. OPTICS EXPRESS,2008,16(25):20345-20353
[31]史广维,张新,张建萍,等.几种失调反射光学系统的波像差特性[J].红外与激光工程,2012,41(1)：160-166
[32]杨新军,王肇圻,母国光,等.偏心和倾斜光学系统的像差特性[J].光子学报,2005,34(11)：1658-1662
[33]Jian W, Banghui G, Qiang S, et al. Third-order aberration fields of pupil decentered optical systems[J]. OPTICS EXPRESS,2012,20(11):11652-11658
[34]张伟,刘剑锋,龙夫年,王治乐.基于Zernike多项式进行波面拟合研究[J].光学技术,2005,31(5)：675-678
[35]黎发志,郑立功,闫锋,薛栋林,张学军.自由曲面的CGH光学检测方法与实验[J].红外与激光工程,2012,41(4)：1052-1056
[36]张文静,刘文广,刘泽金Zemax与Matlab动态数据交换及其应用研究[J].应用光学,2008,29(4)：553-556
[37]J.Michael Rodgers.Four-mirror compact afocal telescope with dual exit pupil[J]. Proc. of SPIE,2006,6342:63421 J(1-7)
[38]齐月静,王平,赵跃进,谢敬辉,糖果果.提高二元计算全息图设计精度的新途径[J].光学技术,2007,33(4)：491-493
[39]康果果,谢敬辉,齐月静,李闻,刘科.用Matlab软件和LCD实现数字全 息图的制作和再现[J].北京理工大学学报,2005,25(6)：509-521
[40]李重阳.用于超高精度非球面面型检测的CGH的设计、制作及误差分析[D]：[硕士学位论文].长春：吉林大学,2013
[41]康果果,谢敬辉,刘科.用于非球面检测的计算全息设计及精度分析[J].光学技术,2007,33(5)：654-657
[42]齐月静,王平,康果果,谢敬辉,刘科,杨辉,赵罘.圆形计算全息图的设计及其参数计算[J].北京理工大学学报,2006,26(9)：821-823
[43]苏萍,谭峭峰,康果果,金国藩.自由曲面零位补偿计算全息图离散相位的B样条拟合[J].光学学报,2010,30(6)：1767-1771
[44]李旭阳,李英才,马臻,等.轻小型CCD相机光学系统设计[J].光子学报,2010,39(6)：994-997
[45]韩昌元.高分辨力空间相机的光学系统研究[J].光学精密工程,2008,16(11)：2164-2172
[46]李旭阳,李英才,马臻,等.高分辨率空间相机共轴三反光学系统实现形式研究[J].应用光学,2009,30(5)：717-723
[47]李旭阳,李英才,马臻,等.消畸变、长焦距相机光学系统设计[J].光学学报,2010,30(9)：2657-2661
[48]Jong U L, Seung M Y. Analytic Design Procedure of Three-mirror Telescope Corrected for Spherical Aberration, Coma, Astigmatism, and Petzval Field Curvature[J]. Journal of the Optical Society of Korea,2009,13(2):184-192
[49]Denis F, Helene D. Manufacturing & Control of the aspherical mirrors for the telescope of the French satellite Pleiades[J]. Proc. of SPIE,2007,6687:66870T(1-11)
[50]M J Sholl, M L Kaplan, M L Lampton. Three Mirror Anastigmat Survey Telescope Optimization [J]. Proc. of SPIE,2008,7010:70103M(1-11)
[51]Robert G, Bruno M, Sebastien V, et al. Zero distortion three mirror telescope designed for the Dark Universe Explorer (DUNE) space mission[J]. Proc. of SPIE, 2006,6265:626549(1-6)
[52]常军,翁志成,姜会林.长焦距空间三反光学系统的设计[J].光学精密工程,2001,9(4)：315-318
[53]樊学武,马卫红,陈荣利,等.具有二次像面的三反射光学系统的研究[J].光 子学报,2003,32(8)：1001-1003
[54]陈浩锋,李英才,樊超,等.宽视场长焦距离轴三反射镜光学系统的设计[J].光子学报,2007,36(SUP)：142-145
[55]常军,姜会林,翁志成,等.大视场、长焦距空间光学系统的设计[J].兵工学报,2003,24(1)：42-44
[56]徐亮,赵建科,周艳,等.长焦距、大视场空间观测相机光学系统设计[J].光学与光电技术,2010,8(6)：64-66
[57]常军,刘莉萍,王涌天,等.大视场、大口径双波段红外非制冷光学系统[J].红外与毫米波学报,2006,25(3)：170-172
[58]姚罡,傅丹膺,黄颖.集成式双线阵立体测绘相机光学系统设计[J].光子学报,2010,39(12)：2229-2233
[59]姚罡,黄颖,傅丹膺.一种易于制造、较大视场离轴三反光学系统设计[J].2010,31(5)：44-48
[60]王美钦,王忠厚,白加光,等.用于高分辨率光谱仪的离轴三反射镜光学系统的设计[J].光电技术应用,2010,25(1)：29-32
[61]杨波,王涌天.自由曲面反射器的计算机辅助设计[J].光学学报,2004,24(6)：721-724
[62]陈云亮,李铁才,邱祥辉.头戴显示器中自由曲面棱镜的设计[J].应用光学,2009,30(4)：552-557
[63]Dewen C, Yongtian W, Hong H, et al. Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism[J]. APPLIED OPTICS,2009,48(14):2655-2668
[64]Dewen C, Yongtian W, Hong H. Free form optical system design with differential equations[J]. Proc. of SPIE,2010,7849:78490Q(1-8)
[65]Ozan C, Gregory E F, Hassan F, et al. Meshfree Approximation Methods for Free-Form Surface Representation in Optical Design with Applications to Head-Worn Displays[J]. Proc. of SPIE,2008,7061:70610D(1-15)
[66]Ozan C, Brendan M, Hassan F, et al. Optimal local shape description for rotationally non-symmetric optical surface design and analysis[J]. OPTICS EXPRESS, 2008,16(3):1583-1589
[67]R A Hicks, Christopher C. Designing coupled free-form surfaces[J]. J. Opt. Soc. Am. A,2010,27(10):2132-2137
[68]Tao M, Jingchi Y, Pei L, et al. Design of a freeform varifocal panoramic optical system with specified annular center of field of view[J]. OPTICS EXPRESS,2011, 19(5):3843-3853
[69]J Buchheister, L Mueller. Reflex free double pass objective due to reflective free form surfaces[J]. Proc. of SPIE,2008,7100:71000N(1-10)
[70]R A Hicks. Direct Methods for Freeform Surface DesignfJ]. Proc. of SPIE,2007, 6668:666802(1-10)
[71]Peter O. Optic design of head-up displays with freeform surfaces specified by NURBS[J]. Proc. of SPIE,2008,7100:71000Y(1-12)
[72]梁敏勇,廖宁放,冯洁,等.三反射式柱面光学系统设计及优化[J].光学学报,2008,28(7)：1359-1363
[73]杨晓飞.三反射镜光学系统的计算机辅助装调技术研究[D]：[博士学位论文].长春：中国科学院长春光学精密机械与物理研究所,2004
[74]赵瑞萍,熊望娥.光学系统计算机辅助装调分析与应用[J].现代电子技术,2010,311：123-132
[75]Yunjong K, Ho-Soon Y, Sug-Whan K, et al. Alignment of off-axis optical system with multi mirrors using derivative of Zernike polynomial coefficient[J]. Proc. of SPIE,2009,7433:74330C(1-8)
[76]Shigeo I, Takeo W, Hiroo K, et al. Development for the alignment procedure of three-aspherical-mirror optics[J]. Proceedings of SPIE,2000,3997:807-813
[77]Seonghui K, Ho-Soon Y, Yun-Woo L, et al. Merit function regression method for efficient alignment control of two-mirror optical systems[J]. OPTICS EXPRESS, 2007,15(8):5059-5068
[78]Eugene D K, Young-Wan C, Myung-Seok K. Reverse-optimization alignment algorithm using Zernike sensitivity [J]. Journal of the Optical Society of Korea,9(2): 68-73
[79]E E Bloemhof, X An, G Kuan, et al. Telescope alignment from sparsely sampled wavefront measurements over pupil subapertures[J]. APPLIED OPTICS,2012,51(3): 394-400
[80]张斌,韩昌元.离轴非球面三反射镜光学系统装调中计算机优化方法的研究[J].光学学报,2001,21(1)：54-58
[81]张斌,张晓辉,韩昌元.光学系统计算机辅助装调中的一种优化算法[J].光学精密工程,2000,8(3)：273-277
[82]杨晓飞,韩昌元.利用离轴三反镜光学系统确定各镜的装调公差[J].光学技术,2005,31(2)：173-176
[83]杨晓飞,张晓辉,韩昌元.用像差逐项优化法装调离轴三反射镜光学系统[J].光学学报,2004,24(1)：115-120
[84]朱时雨,张新,李威.计算机辅助装调与传统基准传递技术相结合实现三镜消像散系统的装调[J].中国光学,2011,4(6)：571-575
[85]史亚莉,高云国,邓伟杰.反射光路的计算机辅助装调模型[J].激光与红外,2009,39(4)：427-430
[86]巩盾,田铁印,王红.利用Zernike系数对离轴三反射系统进行计算机辅助装调[J].光学精密工程,2010,18(8)：1754-1759
[87]孙敬伟,陈涛,王建立,等.主次镜系统的计算机辅助装调[J].光学精密工程,2010,18(10)：2156-2163
[88]梅贵,翟岩,苗健宇.离轴三反式多光谱相机的装调[J].光机电信息,2011,28(8)：1-4
[89]车驰骋,李英才,樊学武,等.基于矢量波像差理论的计算机辅助装调技术研究[J].光子学报,2008,37(8)：1630-1634
[90]孔小辉,樊学武.基于矢量波像差理论的两镜系统装调技术研究[J].激光与光电子学进展,2010,47：082202(1-7)
[91]陈钦芳,李英才,马臻,等.离轴非球面反射镜补偿检验的计算机辅助装调技术研究[J].光子学报,2010,39(12)：2220-2223
[92]李旭阳,李英才,马臻,等.折轴三反射光学系统的计算机辅助装调技术研究[J].应用光学,2009,30(6)：901-906
[93]段学霆,周仁魁,陈建军,等.基于波像差检测的大口径RC光学系统装调分析[J].光子学报,2012,41(1)：36-38
[94]Cao Y, Li L. Novel computer-aided alignment method of optical systems based on genetic algorithm[J]. Proc. of SPIE,2006,6034:60340L(1-7)
[95]Huang Y, Li L. Computer-aided alignment for the segmented mirrors of three-mirror optical system[J]. Proc. of SPIE,2008,6624:66241 V(1-7)
[96]Liu K, Li Y, Liu J, et al. Computer aided alignment of a 20X Schwarzschild projection optics[J]. Proc. of SPIE,2008,7130:713042(1-6)
[97]Yifan H, Lin L. Novel method of computer-aided alignment for large aperture space systems[J]. Proc. of SPIE,2010,7655:76550T(1-6)
[98]林妩媚.光学系统计算机辅助装调机理的研究[J].光电工程,1999,26(Sup.)：49-52
[99]王彬,蒋世磊.卡塞格林系统计算机辅助装调技术研究[J].光学仪器,2008,30(1)：50-54
[100]张伟,刘剑峰,龙夫年.离轴三镜系统光学元件间补偿关系研究[J].光子学报,2005,34(8)：1160-1164
[101]薛晓光,李国喜,龚京忠,等.面向装配过程的精密光学系统计算机辅助装调技术[J].计算机集成制造系统,2011,17(10)：2163-2170
[102]刘琳,王洋,张兴德.基于DWC波前检测的计算机辅助装调技术[J].激光与红外,2013,43(4)：428-432
[103]廖志波,王春雨,栗孟娟,等.透射式光学系统计算机辅助装校技术初步研究[J].红外与激光工程,42(9)：2453-2456
[104]Zhao X, Jiao W, Liao Z, et al. Study on computer-aided alignment method of a three-mirror off-axis aspherical optical system[J].Proc.of SPIE,7656:76566M(1-6)
[2]G.W.Forbes. Characterizing the shape of freeform optics[J]. OPTICS EXPRESS, 2012,20(3):2483-2449
[3]Ozan Cakmakci, Ilhan Kaya, Gregeory E.Fasshauer, Kevin P.Thompson, Jannick P.Rolland. Application of Radial Basis Functions to Represent Optical Freeform Surfacees[J]. Proc. of SPIE,2010,7652:76520A(1-8)
[4]张以谟.应用光学[M].北京：电子工业出版社,2008.494-494
[5]杨波,王涌天.自由曲面反射器的计算机辅助设计[J].光学学报,2004,24(6)：721-724
[6]陈云亮,李铁才,邱祥辉.头戴显示器中自由曲面棱镜的设计[J].应用光学,2009,30(4)：552-557
[7]Dewen C, Yongtian W, Hong H, et al. Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism[J]. APPLIED OPTICS,2009,48(14):2655-2668.
[8]Dewen C, Yongtian W, Hong H. Free form optical system design with differential equations[J]. Proc. of SPIE,2010,7849:78490Q(1-8)
[9]Ozan C, Gregory E F, Hassan F, et al. Meshfree Approximation Methods for Free-Form Surface Representation in Optical Design with Applications to Head-Worn Displays[J]. Proc. of SPIE,2008,7061:70610D(1-15)
[10]Ozan C, Brendan M, Hassan F, et al. Optimal local shape description for rotationally non-symmetric optical surface design and analysis[J]. OPTICS EXPRESS, 2008,16(3):1583-1589.
[11]孙旭涛.应用自由曲面的超薄投影显示系统理论和实验研究[D]：[博士学位论文].杭州：浙江大学,2008
[12]Lei L, Allen Y Y. Design and fabrication of a freeform prism array for 3D microscopy[J]. J. Opt. Soc. Am. A,2010,27(12):2613-2620.
[13]R A Hicks, Christopher C. Designing coupled free-form surfaces[J]. J. Opt. Soc. Am. A,2010,27(10):2132-2137
[14]Tao M, Jingchi Y, Pei L, et al. Design of a freeform varifocal panoramic optical system with specified annular center of field of view[J]. OPTICS EXPRESS,2011, 19(5):3843-3853
[15]J Buchheister, L Mueller. Reflex free double pass objective due to reflective free form surfaces[J]. Proc. of SPIE,2008,7100:71000N(1-10)
[16]R A Hicks. Direct Methods for Freeform Surface Design[J]. Proc. of SPIE,2007, 6668:666802(1-10)
[17]Peter O. Optic design of head-up displays with freeform surfaces specified by NURBS[J]. Proc. of SPIE,2008,7100:71000Y(1-12)
[18]梁敏勇,廖宁放,冯洁,等.三反射式柱面光学系统设计及优化[J].光学学报,2008,28(7)：1359-1363
[19]Kevin T. Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry[J]. J. Opt. Soc. Am. A,2005,22(7):1389-1401
[20]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:spherical aberration [J]. J. Opt. Soc. Am. A,2009,26(5): 1090-1100
[21]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:the astigmatic aberrations [J]. J.Opt.Soc.Am.A,2011,28(5):821-836
[22]Kevin P T. Multinodal fifth-order optical aberrations of optical systems without rotational symmetry:the comatic aberrations[J]. J.Opt.Soc.Am.A,2010,27(6): 1490-1504
[23]Kevin P T, Tobias S, Ozan C. Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry[J]. J. Opt. Soc. Am. A,2009,26(6):1503-1517
[24]John R R. Techniques and tools for obtaining symmetrical performance from tilted-component systems[J]. Optical Engineering,2000,39(7):1776-1787
[25]王灵杰,张新,张建萍,史广维.自由曲面空间光学系统设计[J].2012,33(6)：1040-1046
[26]李庆扬,王能超,易大义.数值分析[M]。北京：清华大学出版社,2008.73-76
[27]王忠仁,张静.复变函数与积分变换[M].北京：高等教育出版社2006.113-116
[28]Tobias S, Kevin P T, Jannick P R. A unique astigmatic nodal property in misaligned Ritchey-Chretien telescopes with misalignment coma removed[J]. OPTICS EXPRESS,2010,18(5):5282-5288
[29]Tobias S, Kevin P T, Jannick P R. Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes[J]. APPLIED OPTICS,49(16):D131-D144
[30]Kevin P T, Tobias S, Jannick P R. The misalignment induced aberrations of TMA telescopes[J]. OPTICS EXPRESS,2008,16(25):20345-20353
[31]史广维,张新,张建萍,等.几种失调反射光学系统的波像差特性[J].红外与激光工程,2012,41(1)：160-166
[32]杨新军,王肇圻,母国光,等.偏心和倾斜光学系统的像差特性[J].光子学报,2005,34(11)：1658-1662
[33]Jian W, Banghui G, Qiang S, et al. Third-order aberration fields of pupil decentered optical systems[J]. OPTICS EXPRESS,2012,20(11):11652-11658
[34]张伟,刘剑锋,龙夫年,王治乐.基于Zernike多项式进行波面拟合研究[J].光学技术,2005,31(5)：675-678
[35]黎发志,郑立功,闫锋,薛栋林,张学军.自由曲面的CGH光学检测方法与实验[J].红外与激光工程,2012,41(4)：1052-1056
[36]张文静,刘文广,刘泽金Zemax与Matlab动态数据交换及其应用研究[J].应用光学,2008,29(4)：553-556
[37]J.Michael Rodgers.Four-mirror compact afocal telescope with dual exit pupil[J]. Proc. of SPIE,2006,6342:63421 J(1-7)
[38]齐月静,王平,赵跃进,谢敬辉,糖果果.提高二元计算全息图设计精度的新途径[J].光学技术,2007,33(4)：491-493
[39]康果果,谢敬辉,齐月静,李闻,刘科.用Matlab软件和LCD实现数字全 息图的制作和再现[J].北京理工大学学报,2005,25(6)：509-521
[40]李重阳.用于超高精度非球面面型检测的CGH的设计、制作及误差分析[D]：[硕士学位论文].长春：吉林大学,2013
[41]康果果,谢敬辉,刘科.用于非球面检测的计算全息设计及精度分析[J].光学技术,2007,33(5)：654-657
[42]齐月静,王平,康果果,谢敬辉,刘科,杨辉,赵罘.圆形计算全息图的设计及其参数计算[J].北京理工大学学报,2006,26(9)：821-823
[43]苏萍,谭峭峰,康果果,金国藩.自由曲面零位补偿计算全息图离散相位的B样条拟合[J].光学学报,2010,30(6)：1767-1771
[44]李旭阳,李英才,马臻,等.轻小型CCD相机光学系统设计[J].光子学报,2010,39(6)：994-997
[45]韩昌元.高分辨力空间相机的光学系统研究[J].光学精密工程,2008,16(11)：2164-2172
[46]李旭阳,李英才,马臻,等.高分辨率空间相机共轴三反光学系统实现形式研究[J].应用光学,2009,30(5)：717-723
[47]李旭阳,李英才,马臻,等.消畸变、长焦距相机光学系统设计[J].光学学报,2010,30(9)：2657-2661
[48]Jong U L, Seung M Y. Analytic Design Procedure of Three-mirror Telescope Corrected for Spherical Aberration, Coma, Astigmatism, and Petzval Field Curvature[J]. Journal of the Optical Society of Korea,2009,13(2):184-192
[49]Denis F, Helene D. Manufacturing & Control of the aspherical mirrors for the telescope of the French satellite Pleiades[J]. Proc. of SPIE,2007,6687:66870T(1-11)
[50]M J Sholl, M L Kaplan, M L Lampton. Three Mirror Anastigmat Survey Telescope Optimization [J]. Proc. of SPIE,2008,7010:70103M(1-11)
[51]Robert G, Bruno M, Sebastien V, et al. Zero distortion three mirror telescope designed for the Dark Universe Explorer (DUNE) space mission[J]. Proc. of SPIE, 2006,6265:626549(1-6)
[52]常军,翁志成,姜会林.长焦距空间三反光学系统的设计[J].光学精密工程,2001,9(4)：315-318
[53]樊学武,马卫红,陈荣利,等.具有二次像面的三反射光学系统的研究[J].光 子学报,2003,32(8)：1001-1003
[54]陈浩锋,李英才,樊超,等.宽视场长焦距离轴三反射镜光学系统的设计[J].光子学报,2007,36(SUP)：142-145
[55]常军,姜会林,翁志成,等.大视场、长焦距空间光学系统的设计[J].兵工学报,2003,24(1)：42-44
[56]徐亮,赵建科,周艳,等.长焦距、大视场空间观测相机光学系统设计[J].光学与光电技术,2010,8(6)：64-66
[57]常军,刘莉萍,王涌天,等.大视场、大口径双波段红外非制冷光学系统[J].红外与毫米波学报,2006,25(3)：170-172
[58]姚罡,傅丹膺,黄颖.集成式双线阵立体测绘相机光学系统设计[J].光子学报,2010,39(12)：2229-2233
[59]姚罡,黄颖,傅丹膺.一种易于制造、较大视场离轴三反光学系统设计[J].2010,31(5)：44-48
[60]王美钦,王忠厚,白加光,等.用于高分辨率光谱仪的离轴三反射镜光学系统的设计[J].光电技术应用,2010,25(1)：29-32
[61]杨波,王涌天.自由曲面反射器的计算机辅助设计[J].光学学报,2004,24(6)：721-724
[62]陈云亮,李铁才,邱祥辉.头戴显示器中自由曲面棱镜的设计[J].应用光学,2009,30(4)：552-557
[63]Dewen C, Yongtian W, Hong H, et al. Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism[J]. APPLIED OPTICS,2009,48(14):2655-2668
[64]Dewen C, Yongtian W, Hong H. Free form optical system design with differential equations[J]. Proc. of SPIE,2010,7849:78490Q(1-8)
[65]Ozan C, Gregory E F, Hassan F, et al. Meshfree Approximation Methods for Free-Form Surface Representation in Optical Design with Applications to Head-Worn Displays[J]. Proc. of SPIE,2008,7061:70610D(1-15)
[66]Ozan C, Brendan M, Hassan F, et al. Optimal local shape description for rotationally non-symmetric optical surface design and analysis[J]. OPTICS EXPRESS, 2008,16(3):1583-1589
[67]R A Hicks, Christopher C. Designing coupled free-form surfaces[J]. J. Opt. Soc. Am. A,2010,27(10):2132-2137
[68]Tao M, Jingchi Y, Pei L, et al. Design of a freeform varifocal panoramic optical system with specified annular center of field of view[J]. OPTICS EXPRESS,2011, 19(5):3843-3853
[69]J Buchheister, L Mueller. Reflex free double pass objective due to reflective free form surfaces[J]. Proc. of SPIE,2008,7100:71000N(1-10)
[70]R A Hicks. Direct Methods for Freeform Surface DesignfJ]. Proc. of SPIE,2007, 6668:666802(1-10)
[71]Peter O. Optic design of head-up displays with freeform surfaces specified by NURBS[J]. Proc. of SPIE,2008,7100:71000Y(1-12)
[72]梁敏勇,廖宁放,冯洁,等.三反射式柱面光学系统设计及优化[J].光学学报,2008,28(7)：1359-1363
[73]杨晓飞.三反射镜光学系统的计算机辅助装调技术研究[D]：[博士学位论文].长春：中国科学院长春光学精密机械与物理研究所,2004
[74]赵瑞萍,熊望娥.光学系统计算机辅助装调分析与应用[J].现代电子技术,2010,311：123-132
[75]Yunjong K, Ho-Soon Y, Sug-Whan K, et al. Alignment of off-axis optical system with multi mirrors using derivative of Zernike polynomial coefficient[J]. Proc. of SPIE,2009,7433:74330C(1-8)
[76]Shigeo I, Takeo W, Hiroo K, et al. Development for the alignment procedure of three-aspherical-mirror optics[J]. Proceedings of SPIE,2000,3997:807-813
[77]Seonghui K, Ho-Soon Y, Yun-Woo L, et al. Merit function regression method for efficient alignment control of two-mirror optical systems[J]. OPTICS EXPRESS, 2007,15(8):5059-5068
[78]Eugene D K, Young-Wan C, Myung-Seok K. Reverse-optimization alignment algorithm using Zernike sensitivity [J]. Journal of the Optical Society of Korea,9(2): 68-73
[79]E E Bloemhof, X An, G Kuan, et al. Telescope alignment from sparsely sampled wavefront measurements over pupil subapertures[J]. APPLIED OPTICS,2012,51(3): 394-400
[80]张斌,韩昌元.离轴非球面三反射镜光学系统装调中计算机优化方法的研究[J].光学学报,2001,21(1)：54-58
[81]张斌,张晓辉,韩昌元.光学系统计算机辅助装调中的一种优化算法[J].光学精密工程,2000,8(3)：273-277
[82]杨晓飞,韩昌元.利用离轴三反镜光学系统确定各镜的装调公差[J].光学技术,2005,31(2)：173-176
[83]杨晓飞,张晓辉,韩昌元.用像差逐项优化法装调离轴三反射镜光学系统[J].光学学报,2004,24(1)：115-120
[84]朱时雨,张新,李威.计算机辅助装调与传统基准传递技术相结合实现三镜消像散系统的装调[J].中国光学,2011,4(6)：571-575
[85]史亚莉,高云国,邓伟杰.反射光路的计算机辅助装调模型[J].激光与红外,2009,39(4)：427-430
[86]巩盾,田铁印,王红.利用Zernike系数对离轴三反射系统进行计算机辅助装调[J].光学精密工程,2010,18(8)：1754-1759
[87]孙敬伟,陈涛,王建立,等.主次镜系统的计算机辅助装调[J].光学精密工程,2010,18(10)：2156-2163
[88]梅贵,翟岩,苗健宇.离轴三反式多光谱相机的装调[J].光机电信息,2011,28(8)：1-4
[89]车驰骋,李英才,樊学武,等.基于矢量波像差理论的计算机辅助装调技术研究[J].光子学报,2008,37(8)：1630-1634
[90]孔小辉,樊学武.基于矢量波像差理论的两镜系统装调技术研究[J].激光与光电子学进展,2010,47：082202(1-7)
[91]陈钦芳,李英才,马臻,等.离轴非球面反射镜补偿检验的计算机辅助装调技术研究[J].光子学报,2010,39(12)：2220-2223
[92]李旭阳,李英才,马臻,等.折轴三反射光学系统的计算机辅助装调技术研究[J].应用光学,2009,30(6)：901-906
[93]段学霆,周仁魁,陈建军,等.基于波像差检测的大口径RC光学系统装调分析[J].光子学报,2012,41(1)：36-38
[94]Cao Y, Li L. Novel computer-aided alignment method of optical systems based on genetic algorithm[J]. Proc. of SPIE,2006,6034:60340L(1-7)
[95]Huang Y, Li L. Computer-aided alignment for the segmented mirrors of three-mirror optical system[J]. Proc. of SPIE,2008,6624:66241 V(1-7)
[96]Liu K, Li Y, Liu J, et al. Computer aided alignment of a 20X Schwarzschild projection optics[J]. Proc. of SPIE,2008,7130:713042(1-6)
[97]Yifan H, Lin L. Novel method of computer-aided alignment for large aperture space systems[J]. Proc. of SPIE,2010,7655:76550T(1-6)
[98]林妩媚.光学系统计算机辅助装调机理的研究[J].光电工程,1999,26(Sup.)：49-52
[99]王彬,蒋世磊.卡塞格林系统计算机辅助装调技术研究[J].光学仪器,2008,30(1)：50-54
[100]张伟,刘剑峰,龙夫年.离轴三镜系统光学元件间补偿关系研究[J].光子学报,2005,34(8)：1160-1164
[101]薛晓光,李国喜,龚京忠,等.面向装配过程的精密光学系统计算机辅助装调技术[J].计算机集成制造系统,2011,17(10)：2163-2170
[102]刘琳,王洋,张兴德.基于DWC波前检测的计算机辅助装调技术[J].激光与红外,2013,43(4)：428-432
[103]廖志波,王春雨,栗孟娟,等.透射式光学系统计算机辅助装校技术初步研究[J].红外与激光工程,42(9)：2453-2456
[104]Zhao X, Jiao W, Liao Z, et al. Study on computer-aided alignment method of a three-mirror off-axis aspherical optical system[J].Proc.of SPIE,7656:76566M(1-6)