基于畸变校正的成像自由曲面光学设计
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摘要
随着人们对光学系统轻量化、集成化以及新颖性等要求的提高,传统的光学元器件已经不能满足现代光学系统的需求了。自由曲面由于其大的自由度,能够按照人为设定的要求控制光线的出射方向以及照度的分布等,将它用于光学系统中是近年来光学设计领域很大的一个趋势。
本文从非平面成像的新颖性、大的市场需求性等出发,在分析非平面成像与传统平面成像存在很大的不同、用传统的镜头成像会产生很大的视觉畸变的基础上,总结了非平面成像现有的畸变校正方法的优缺点,提出了用自由曲面校正非平面成像视觉畸变的可能性。
首先本文分析了非平面或者复杂表面上成像畸变的产生原因以及视觉畸变校正所基于的光学原理,提出了一种用于计算观察在有限距离的畸变计算方法。在已知原始成像镜头的基础上,在球坐标系下,考虑每一个视场的主光线,利用折射定律,建立待求的自由曲面上每一点的矢径长度、矢径与z轴的夹角以及矢径在xoy平面上投影线段与x轴的夹角这三个量与物(像)点位置之间关系的偏微分方程,提出了基于偏微分方程的自由曲面设计方法。用差分法得到方程的数值解,即可得到自由曲面上的数据点信息。利用这种方法实现了球面投影系统,其畸变被校正在很小的范围内,成像质量基本上符合投影系统的观察要求。
鉴于基于偏微分方程的自由曲面设计方法中,对每个视场的成像光线仅仅考虑主光线,在校正畸变的同时会影响成像质量这一缺点,本文还提出了一种基于微球面的自由曲面设计方法。对应每个成像的视场点,考虑经过子午平面上出瞳边缘的两根光线和主光线这三根光线,用一个微球面透镜(两个微球面)实现每个视场点的三根光线到其像点的会聚。把通过这种方法设计的透镜用在球面投影系统中,与偏微分方程方法的结果进行比较,这种方法在校正畸变的同时,能够有效地控制成像质量的恶化。
随着基于微球面的自由曲面设计方法研究的深入,我们发现,每个视场计算子午平面的三根成像光线能够有效地提高成像质量,但是其弧矢方向的成像质量提高远不如子午方向明显,这是因为我们计算的成像光线都在子午平面里,弧矢平面的成像光线到其像点的会聚并不被控制。所以,接下来,我们改进了基于微球面的自由曲面设计方法,对于每个成像的视场点,分别计算出瞳子午平面一条边缘光线、出瞳弧矢平面上一条边缘光线以及主光线,这种计算方法必将把算法扩展到三维,使得这种方法具有更普遍的适用性。本文还利用这种扩展的基于微球面的自由曲面设计方法设计了一个用于柱面投影的自由曲面透镜,这是一个非旋转对称的系统,不管是在畸变还是成像质量上都达到了预期的要求。
最后,本文对自由曲面用于成像进行了总结和展望。本文中提出的自由曲面设计方法是基于非平面成像畸变校正的,自由曲面还能用于其它各种类型的光学系统。在未来的研究中,自由曲面主要在面型描述方式、精确的像差计算方法以及优化效率和光线追迹效率等方面。
With the demands of lightweight, compactness and novelty of the modern optical system, traditional optical elements like spherical and aspherical surfaces cannot meet the needs any more. Freeform surface has larger degrees of freedom, so it can control the illumination distribution and the outgoing direction of every ray as needed. Using freeform surface in optical system is a big trend in the recent years.
In this paper, we started from the novelty and market demand of the non-planar imaging system. We aware that use ordinary imaging lens in non-planar imaging system will cause large distortion. Based on the analysis of the existing distortion correction methods, we propose that there is possibility that we can correct the non-planar imaging distortion using freeform surface.
First of all, this paper analyzed the reason of the distortion on non-planar surface. Based on the visual principle of distortion correction, a new relationship between object points and image points is proposed. Employing the Snell's law on the chief ray of each field of view, freeform design method based on partial differential equation was proposed. Based on the original imaging lens, each chief ray was reflected to their new imaging point, so the visual distortion can be corrected. Using this method, a freeform lens was designed to realize spherical projection. The distortion of the system was low enough to observe, and the image quality was high enough for projection system but not as high as the original one.
In partial differential equation method, only the chief ray of each field of view was considered. This may correct the distortion while at the same time influence the image quality. Then this paper proposed another design method, which can design two rotationally symmetric freeform surfaces simultaneously, based on the micro spheres. In this method, we realize three rays (one chief ray and two marginal rays in the tangential plane) of each field of view reflecting to the new image point by using a macro spherical lens. Employing the middle point of each macro sphere, the profiles of the freeform surface can be obtained. For comparability with the partial differential method, a freeform lens with two freeform surfaces was designed for spherical projection using this method. Not only the distortion was corrected well but also the image quality was better than that in the partial differential equation method.
With the deep study of the macro sphere method, we found that consider two marginal rays both from the tangential plane will cause a problem, which was the image quality of the sagittal plane was much lower than the image quality of the tangential plane. That is because the rays from the sagittal plane did not focused as well as that in tangential plane. Then we proposed a design method for non-rotationally symmetric surface based on the macro sphere by considering one marginal ray from the tangential plane, one marginal ray from the sagittal plane and also a chief ray. This extends the macro sphere method to3D, and makes it more general. Then using this method, a non-rotationally symmetric freeform lens was design for cylinder projection system. Both the distortion and the image quality of the system have satisfied our need.
At last, this paper put a summary and outlook on freeform lens design and the non-planar imaging. The freeform design methods proposed in this paper aimed to distortion correction of the non-planar imaging, but the freeform lens had much wider application. The research of freeform surface will focus on better description method, more precise imaging aberration calculation, higher ray tracing efficiency and optimization efficiency in the near future.
本文从非平面成像的新颖性、大的市场需求性等出发,在分析非平面成像与传统平面成像存在很大的不同、用传统的镜头成像会产生很大的视觉畸变的基础上,总结了非平面成像现有的畸变校正方法的优缺点,提出了用自由曲面校正非平面成像视觉畸变的可能性。
首先本文分析了非平面或者复杂表面上成像畸变的产生原因以及视觉畸变校正所基于的光学原理,提出了一种用于计算观察在有限距离的畸变计算方法。在已知原始成像镜头的基础上,在球坐标系下,考虑每一个视场的主光线,利用折射定律,建立待求的自由曲面上每一点的矢径长度、矢径与z轴的夹角以及矢径在xoy平面上投影线段与x轴的夹角这三个量与物(像)点位置之间关系的偏微分方程,提出了基于偏微分方程的自由曲面设计方法。用差分法得到方程的数值解,即可得到自由曲面上的数据点信息。利用这种方法实现了球面投影系统,其畸变被校正在很小的范围内,成像质量基本上符合投影系统的观察要求。
鉴于基于偏微分方程的自由曲面设计方法中,对每个视场的成像光线仅仅考虑主光线,在校正畸变的同时会影响成像质量这一缺点,本文还提出了一种基于微球面的自由曲面设计方法。对应每个成像的视场点,考虑经过子午平面上出瞳边缘的两根光线和主光线这三根光线,用一个微球面透镜(两个微球面)实现每个视场点的三根光线到其像点的会聚。把通过这种方法设计的透镜用在球面投影系统中,与偏微分方程方法的结果进行比较,这种方法在校正畸变的同时,能够有效地控制成像质量的恶化。
随着基于微球面的自由曲面设计方法研究的深入,我们发现,每个视场计算子午平面的三根成像光线能够有效地提高成像质量,但是其弧矢方向的成像质量提高远不如子午方向明显,这是因为我们计算的成像光线都在子午平面里,弧矢平面的成像光线到其像点的会聚并不被控制。所以,接下来,我们改进了基于微球面的自由曲面设计方法,对于每个成像的视场点,分别计算出瞳子午平面一条边缘光线、出瞳弧矢平面上一条边缘光线以及主光线,这种计算方法必将把算法扩展到三维,使得这种方法具有更普遍的适用性。本文还利用这种扩展的基于微球面的自由曲面设计方法设计了一个用于柱面投影的自由曲面透镜,这是一个非旋转对称的系统,不管是在畸变还是成像质量上都达到了预期的要求。
最后,本文对自由曲面用于成像进行了总结和展望。本文中提出的自由曲面设计方法是基于非平面成像畸变校正的,自由曲面还能用于其它各种类型的光学系统。在未来的研究中,自由曲面主要在面型描述方式、精确的像差计算方法以及优化效率和光线追迹效率等方面。
With the demands of lightweight, compactness and novelty of the modern optical system, traditional optical elements like spherical and aspherical surfaces cannot meet the needs any more. Freeform surface has larger degrees of freedom, so it can control the illumination distribution and the outgoing direction of every ray as needed. Using freeform surface in optical system is a big trend in the recent years.
In this paper, we started from the novelty and market demand of the non-planar imaging system. We aware that use ordinary imaging lens in non-planar imaging system will cause large distortion. Based on the analysis of the existing distortion correction methods, we propose that there is possibility that we can correct the non-planar imaging distortion using freeform surface.
First of all, this paper analyzed the reason of the distortion on non-planar surface. Based on the visual principle of distortion correction, a new relationship between object points and image points is proposed. Employing the Snell's law on the chief ray of each field of view, freeform design method based on partial differential equation was proposed. Based on the original imaging lens, each chief ray was reflected to their new imaging point, so the visual distortion can be corrected. Using this method, a freeform lens was designed to realize spherical projection. The distortion of the system was low enough to observe, and the image quality was high enough for projection system but not as high as the original one.
In partial differential equation method, only the chief ray of each field of view was considered. This may correct the distortion while at the same time influence the image quality. Then this paper proposed another design method, which can design two rotationally symmetric freeform surfaces simultaneously, based on the micro spheres. In this method, we realize three rays (one chief ray and two marginal rays in the tangential plane) of each field of view reflecting to the new image point by using a macro spherical lens. Employing the middle point of each macro sphere, the profiles of the freeform surface can be obtained. For comparability with the partial differential method, a freeform lens with two freeform surfaces was designed for spherical projection using this method. Not only the distortion was corrected well but also the image quality was better than that in the partial differential equation method.
With the deep study of the macro sphere method, we found that consider two marginal rays both from the tangential plane will cause a problem, which was the image quality of the sagittal plane was much lower than the image quality of the tangential plane. That is because the rays from the sagittal plane did not focused as well as that in tangential plane. Then we proposed a design method for non-rotationally symmetric surface based on the macro sphere by considering one marginal ray from the tangential plane, one marginal ray from the sagittal plane and also a chief ray. This extends the macro sphere method to3D, and makes it more general. Then using this method, a non-rotationally symmetric freeform lens was design for cylinder projection system. Both the distortion and the image quality of the system have satisfied our need.
At last, this paper put a summary and outlook on freeform lens design and the non-planar imaging. The freeform design methods proposed in this paper aimed to distortion correction of the non-planar imaging, but the freeform lens had much wider application. The research of freeform surface will focus on better description method, more precise imaging aberration calculation, higher ray tracing efficiency and optimization efficiency in the near future.
引文
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5丁毅,顾培夫,陆巍,郑臻荣,利用偏微分方程数值解构造自由曲面反光器,浙江大学学报(工学版),41(9)：1516-1518,2007.
6丁毅,顾培夫,实现均匀照明的自由曲面反射器,光学学报,27(3)：540-544,2007.
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8 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications 285(6):986-991,2012.
9 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction Using a Freeform Lens for Projection onto a Non-Planar Surface, Society for Information Display 2012 International Symposium Digest,910-913, 2012.
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9 Fabian Duerr, Pablo Benitez,Juan C. Minano,Youri Meuretand Hugo Thienpont, Analytic design method for optimal imaging:coupling three ray sets using two free-form lens profiles, OPTICS EXPRESS 20:5576-5585, 2012.
10李晓彤,岑兆丰,几何光学·像差·光学设计,浙江大学出版社,2003.
11丁毅,自由曲面光学器件的设计及其在照明系统中的应用,[博士学位论文].杭州,浙江大学,2009.
12 Rengmao Wu, Zhenrong Zheng, Haifeng Li. and Xu Liu, Constructing optical freeform surfaces using unit tangent vectors of feature data points, J. Opt. Soc. Am. A, Vol.28(9):1880-1888,2011.
13吴仍茂,自由曲面照明设计方法的研究,[博士学位论文].杭州,浙江大学,2013.
14http://zh.wikipedia.org/wiki/%E9%9D%9E/E5%9D%87%E5%8C%80%E6%9C%89%E7%90%86B%E6 %AO%B7%E6%9D%A1
15 Les Piegl, Wayne Tiller著,赵罡,穆国旺,王拉柱译,非均匀有理B样条,清华大学出版社,2010.
16 Wolfgang Bohm, Gerald E. Farin, Jurgen Kahmann, A survey of curve and surface methods in CAGD, Computer Aided Geometric Design 1(1):1-60,1984.
17 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications 285(6):986-991,2012.
18 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction Using a Freeform Lens for Projection onto a Non-Planar Surface, Society for Information Display 2012 International Symposium Digest, 910-913,2012.
19 Jia Hou, Haifeng Li, Rengmao Wu, Peng Liu, Zhenrong Zheng, Xu Liu, A method to design two aspheric surfaces for imaging system, Applied Optics 52(11):2294-2299,2013.
20刘旭、李海峰,现代投影显示技术,浙江大学出版社,2009.
1 Teus W. Tukker, Beam-shaping lenses in illumination optics, Proc. of SPIE vol.6338:63380A(1-5),2006.
2 Pablo Benitez, Juan C. Minano, Jose Blen, Ruben Mohedano, Ju'lio Chaves, Oliver Dross, Maikel Herna'ndez, Waqidi Falicoff, Simultaneous multiple surface optical design method in three dimensions, Opt. Eng. vol.43(7):1489-1502,2004.
3 P. Benitez, J.C. Minano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, J.L. Alvarez, W. Falicoff, SMS Design Method in 3D Geometry:Examples and Applications, Proceedings of SPIE vol.5185: 18-29.
4 Juan C. Minano and Pablo Benitez, SMS design method 3D:Calculating multiple freeform surfaces from the optical prescription, Frontiers in Optics 2003, OSA Technical Digest,2003.
5 Manuel Gutierrez, Juan C. Minnano, Carlos Vega, and Pablo Benitez, Application of Lorentz geometry to nonimaging optics:new 3D ideal concentrators, J. Opt. Soc. Am. A 13(3):532-540,1996.
6 H. Ries and J. Muschaweck, Tailored freeform optical surfaces, J. Opt. Soc. Am. A vol.19(3):590-595, 2002.
7 Kevin P. Thompson and Jannick P. Rolland, Freeform Optical Surfaces:A Revolution in Imaging Optical Design, Optics & Photonics News,30-35,2012.
8 Axel B'auerle, Adrien Bruneton. Rolf Wester, Jochen Stollenwerk, and Peter Loosen, Algorithm for irradiance tailoring using multiple freeform optical surfaces, OPTICS EXPRESS vol.20(13):14477-14485, 2012.
9 Lei Li and Allen Y. Yi, Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera, Applied Optics vol.51(12):1843-1852,2012.
1 R. Andrew Hicks, Catadioptric Sensors that Approximate Wide-angle Perspective Projections, IEEE Vol.1063-6919/00:1-7,2000.
2 R. Andrew Hicks, Direct Methods for Freeform Surface Design, Proc. of SPIE Vol.6668 (666802):1-10, 2007.
3 Yi Ding, Xu Liu, Zhenrong Zheng, Peifu Gu, Freeform LED lens for uniform illumination, OPTICS EXPRESS Vol.16(17):12958-12966,2008.
4 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications Vol.285(6):986-991,2012.
5 Dewen Cheng, Yongtian Wang, Hong Hua, Free form optical system design with differential equations, Proc. of SPIE Vol.7849(78490Q):1-8,2011.
6 Dejan Graboviokic, Pablo Benitez and Juan C. Minano, Aspheric V-groove reflector design with the SMS method in two dimensions, OPTICS EXPRESS Vol.18(3):2515-2521,2010.
7 Juan C. Minano, Pablo Benitez, Wang Lin, Fernando Munoz, Jose Infante and Asuncion Santamariaa, Overview of the SMS design method applied to imaging optics, Proc. of SPIE Vol.7429,74290C:1-9,2009.
8 Fabian Duerr, Pablo Benitez.Juan C. Minano,Youri Meuretand Hugo Thienpont, Analytic design method for optimal imaging:coupling three ray sets using two free-form lens profiles, OPTICS EXPRESS Vol.20(5): 5576-5585,2012.
9 Yi Luo, Zexin Feng, Yanjun Han, Hongtao Li, Design of compact and smooth free-form optical system with uniform illuminance for LED source, OPTICS EXPRESS Vol.18(9):9055-9063,2010.
10 Rengmao Wu, Zhenrong Zheng, Haifeng Li, and Xu Liu, Optimization design of irradiance array for LED uniform rectangular illumination, Vol.51(13):2257-2263,2012.
11 Li Xu, Kexin Chen, Qingsheng He, Guofan Jin, Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism, Vol.(48)15:2871-2879,2009.
12 Xutao Sun, Zhenrong Zheng, Xu Liu, Peifu Gu, Etendue analysis and measurement of light source with elliptical reflector, Displays 27, pp56-61,2006.
13 Zhenrong Zheng, Design of off-axis reflective projection lens using spherical Fresnel surface, Optik 122, pp145-149,2011.
14孙旭涛,郑臻荣,刘旭,顾培夫,用Zernikc自由曲面设计弯曲屏幕超薄投影系统,浙江大学学报工学版,Vol.8,1428-1432,2009.
15程德文,王涌天,常军,刘越,徐况,轻型大视场自由曲面棱镜头盔显示器的设计,红外与激光工 程,Vo1.36(3)：309-313,2007.
16 Dewen Cheng, Yongtian Wang, Hong Hua, M. M. Talha, Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism, APPLIED OPTICS, Vol.48(14): 2655-2668,2009.
17 Dewen Cheng, Qingfeng Wang, Yongtian Wang, Guofan Jin, Lightweight spatial-multiplexed dual focal-plane head-mounted display using two freeform prisms, CHINESE OPTICS LETTERS Vol.11(3): 031201-1-4,2013.
18 Dewen Cheng, Yongtian Wang, Hong Hua, and Jose Sasian, Design of a wide-angle, lightweight head-mounted display using free-form optics tiling, OPTICS LETTERS Vol.36(11):2098-2100,2011.
19 Jannick P. Rolland and Kevin P. Thompson, Freeform optics:Evolution? No, revolutio! SPIE Newsroom 10.1117/2.1201207.004309, pp1-3,2012.
20王涌天,自由曲面光学系统设计及其应用,光学与光电技术,Vol.10(3)：14-17,2012.
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22程灏波,冯云鹏,王涌天,自由曲面光学研究,专题报告光学制造,pp17-22,2009.
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21 Http://www.opticinnovations.co.uk/attachments/article with pdf attachments/Oiotiq%20-%20Commercial %20relevance%20of%20free%20form%20optics.pdf
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31 Dewen Cheng, Yongtian Wang, Hong Hua, Free form optical system design with differential equations, Proc. of SPIE Vol.7849(78490Q):1-8,2011.
32 Yi Luo, Zexin Feng, Yanjun Han, Hongtao Li, Design of compact and smooth free-form optical system with uniform illuminance for LED source, OPTICS EXPRESS Vol.18(9):9055-9063,2010.
33 Rengmao Wu, Zhenrong Zheng, Haifeng Li, and Xu Liu, Optimization design of irradiance array for LED uniform rectangular illumination, Vol.51(13):2257-2263,2012.
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36 R. Andrew Hicks, Direct Methods for Freeform Surface Design, Proc. of SPIE vol.6668 (666802):1-10, 2007.
37. Dejan Grabovickic, Pablo Benitez and Juan C. Minano, Aspheric V-groove reflector design with the SMS method in two dimensions, OPTICS EXPRESS 18(3):2515-2521,2010.
38. Juan C. Minano, Pablo Benitez, Wang Lin, Fernando Munoz, Jose Infante and Asuncion Santamariaa, Overview of the SMS design method applied to imaging optics, Proc. of SPIE 7429,74290C:1-9,2009.
39. Fabian Duerr, Pablo Benitez,Juan C. Minano,Youri Meuretand Hugo Thienpont, Analytic design method for optimal imaging:coupling three ray sets using two free-form lens profiles, OPTICS EXPRESS 20(5): 5576-5585,2012.
40.侯佳,自由曲面光学设计理论,《10000个科学难题-信息科学卷》,pp929-931,科学出版社,2011.
41. Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu. Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications 285(6):986-991,2012.
42. Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction Using a Freeform Lens for Projection onto a Non-Planar Surface, Society for Information Display 2012 International Symposium Digest, 910-913,2012.
43 Jia Hou, Haifeng Li, Rengmao Wu, Peng Liu, Zhenrong Zheng, Xu Liu, A method to design two aspheric surfaces for imaging system, Applied Optics 52(11):2294-2299,2013.
44.张坤领,林彬,王晓峰,非球面加工现状,组合机床与自动化加工技术,专题评述(5)：1-5,2007.
45.丁毅,自由曲面光学器件的设计及其在照明系统中的应用,[博士学位论文].杭州,浙江大学,2009.
46.李晓彤,岑兆丰,几何光学·像差·光学设计,浙江大学出版社,2003.
47. R. Andrew Hicks, Direct Methods for Freeform Surface Design, invited paper, Proc. of SPIE Vol.6668 666802:1-10,2007.
48 Dewen Cheng, Yongtian Wang, Hong Hua, M. M. Talha, Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism, APPLIED OPTICS, Vol.48(14): 2655-2668,2009.
49 http://optics.nasa.gov/tech days/tech days 2011/Day%202%20Wed%20June%2022/29%20UoR%20Path %20to%20Freeform%20Optics.pdf.
1 Benjamin A. Jacobson and Robert D Gengelbach, Lens for uniform LED illumination:an example of automated optimization using Monte Carlo ray-tracing of an LED source. Proc. of SPIE,4446:130-138,2002.
2 Steven Doyle, David Corcoran and Jon Connell, Automated mirror design for an extended light source, Proc. of SPIE,3781:94-102,1999.
3 John Bortz, Narkis Shatz and David Pitou, Optimal design of a nonimaging projection lens for use with an LED source and a rectangular target, Proc. of SPIE,4092:130-138,2000.
4 Thomas L. R. Davenport,3D UNRBS representation of surfaces for illumination. Proc. of SPIE,4832: 293-301,2002.
5丁毅,顾培夫,陆巍,郑臻荣,利用偏微分方程数值解构造自由曲面反光器,浙江大学学报(工学版),41(9)：1516-1518,2007.
6丁毅,顾培夫,实现均匀照明的自由曲面反射器,光学学报,27(3)：540-544,2007.
7丁毅,自由曲面光学器件的设计及其在照明系统中的应用,[博士学位论文].杭州,浙江大学,2009.
8 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications 285(6):986-991,2012.
9 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction Using a Freeform Lens for Projection onto a Non-Planar Surface, Society for Information Display 2012 International Symposium Digest,910-913, 2012.
10李晓彤,岑兆丰,几何光学·像差·光学设计,浙江大学出版社,2003.
11 Harald Ries, Tailored freeform optical surfaces, Optical Society of America,19(3):590-595,2002.
12陈维恒,微分几何,北京大学出版社,2006.
13孙鑫,白加光,王忠厚,球幕投影光学系统的设计与研究,光学学报,2006,35(11)：1766-1770.
14孙鑫,飞行模拟用小型球幕投影系统的研究,中国科学院研究生院硕士论文,2006.
1 G. D. Wassermann and E. Wolf, On the theory of aplanatic aspheric systems, Proc. Phys. Soc. B 62:2-8, 1949.
2 Kenro Miyamoyo, On the Design of Optical Systems with an Aspheric Surface, J. Opt. Soc. Am.51:21-22, 1961.
3 Teus W. Tukker, Beam-shaping lenses in illumination optics, Proc. of SPIE 6338,63380A:1-5,2006.
4 R. Andrew Hicks and Christopher Croke, Designing coupled free-form surfaces, J. Opt. Soc. Am. A 27(10): 2132-2137.
5 Juan C. Minano, Pablo Benitez, Wang Lin, Jose Infante, Fernando Munoz and Asuncion Santamaria, An application of the SMS method for imaging designs, OPTICS EXPRESS 17:24036-24044,2009.
6 Pablo Benitez and Juan C. Minano, Ultrahigh-numerical-aperture imaging concentrator, J. Opt. Soc. Am. A 14:1988-1997,1997.
7 Dejan Grabovickic, Pablo Benitez and Juan C. Minano, Aspheric V-groove reflector design with the SMS method in two dimensions, OPTICS EXPRESS 18:2515-2521,2010.
8 Juan C. Minano, Pablo Benitez, Wang Lin, Fernando Munoz, Jose Infante and Asuncion Santamariaa, Overview of the SMS design method applied to imaging optics, Proc. of SPIE 7429:74290C1-9,2009.
9 Fabian Duerr, Pablo Benitez,Juan C. Minano,Youri Meuretand Hugo Thienpont, Analytic design method for optimal imaging:coupling three ray sets using two free-form lens profiles, OPTICS EXPRESS 20:5576-5585, 2012.
10李晓彤,岑兆丰,几何光学·像差·光学设计,浙江大学出版社,2003.
11丁毅,自由曲面光学器件的设计及其在照明系统中的应用,[博士学位论文].杭州,浙江大学,2009.
12 Rengmao Wu, Zhenrong Zheng, Haifeng Li. and Xu Liu, Constructing optical freeform surfaces using unit tangent vectors of feature data points, J. Opt. Soc. Am. A, Vol.28(9):1880-1888,2011.
13吴仍茂,自由曲面照明设计方法的研究,[博士学位论文].杭州,浙江大学,2013.
14http://zh.wikipedia.org/wiki/%E9%9D%9E/E5%9D%87%E5%8C%80%E6%9C%89%E7%90%86B%E6 %AO%B7%E6%9D%A1
15 Les Piegl, Wayne Tiller著,赵罡,穆国旺,王拉柱译,非均匀有理B样条,清华大学出版社,2010.
16 Wolfgang Bohm, Gerald E. Farin, Jurgen Kahmann, A survey of curve and surface methods in CAGD, Computer Aided Geometric Design 1(1):1-60,1984.
17 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications 285(6):986-991,2012.
18 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction Using a Freeform Lens for Projection onto a Non-Planar Surface, Society for Information Display 2012 International Symposium Digest, 910-913,2012.
19 Jia Hou, Haifeng Li, Rengmao Wu, Peng Liu, Zhenrong Zheng, Xu Liu, A method to design two aspheric surfaces for imaging system, Applied Optics 52(11):2294-2299,2013.
20刘旭、李海峰,现代投影显示技术,浙江大学出版社,2009.
1 Teus W. Tukker, Beam-shaping lenses in illumination optics, Proc. of SPIE vol.6338:63380A(1-5),2006.
2 Pablo Benitez, Juan C. Minano, Jose Blen, Ruben Mohedano, Ju'lio Chaves, Oliver Dross, Maikel Herna'ndez, Waqidi Falicoff, Simultaneous multiple surface optical design method in three dimensions, Opt. Eng. vol.43(7):1489-1502,2004.
3 P. Benitez, J.C. Minano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, J.L. Alvarez, W. Falicoff, SMS Design Method in 3D Geometry:Examples and Applications, Proceedings of SPIE vol.5185: 18-29.
4 Juan C. Minano and Pablo Benitez, SMS design method 3D:Calculating multiple freeform surfaces from the optical prescription, Frontiers in Optics 2003, OSA Technical Digest,2003.
5 Manuel Gutierrez, Juan C. Minnano, Carlos Vega, and Pablo Benitez, Application of Lorentz geometry to nonimaging optics:new 3D ideal concentrators, J. Opt. Soc. Am. A 13(3):532-540,1996.
6 H. Ries and J. Muschaweck, Tailored freeform optical surfaces, J. Opt. Soc. Am. A vol.19(3):590-595, 2002.
7 Kevin P. Thompson and Jannick P. Rolland, Freeform Optical Surfaces:A Revolution in Imaging Optical Design, Optics & Photonics News,30-35,2012.
8 Axel B'auerle, Adrien Bruneton. Rolf Wester, Jochen Stollenwerk, and Peter Loosen, Algorithm for irradiance tailoring using multiple freeform optical surfaces, OPTICS EXPRESS vol.20(13):14477-14485, 2012.
9 Lei Li and Allen Y. Yi, Design and fabrication of a freeform microlens array for a compact large-field-of-view compound-eye camera, Applied Optics vol.51(12):1843-1852,2012.
1 R. Andrew Hicks, Catadioptric Sensors that Approximate Wide-angle Perspective Projections, IEEE Vol.1063-6919/00:1-7,2000.
2 R. Andrew Hicks, Direct Methods for Freeform Surface Design, Proc. of SPIE Vol.6668 (666802):1-10, 2007.
3 Yi Ding, Xu Liu, Zhenrong Zheng, Peifu Gu, Freeform LED lens for uniform illumination, OPTICS EXPRESS Vol.16(17):12958-12966,2008.
4 Jia Hou, Haifeng Li, Zhenrong Zheng, Xu Liu, Distortion Correction for Imaging on Non-planar Surface Using Freeform Lens, Optics Communications Vol.285(6):986-991,2012.
5 Dewen Cheng, Yongtian Wang, Hong Hua, Free form optical system design with differential equations, Proc. of SPIE Vol.7849(78490Q):1-8,2011.
6 Dejan Graboviokic, Pablo Benitez and Juan C. Minano, Aspheric V-groove reflector design with the SMS method in two dimensions, OPTICS EXPRESS Vol.18(3):2515-2521,2010.
7 Juan C. Minano, Pablo Benitez, Wang Lin, Fernando Munoz, Jose Infante and Asuncion Santamariaa, Overview of the SMS design method applied to imaging optics, Proc. of SPIE Vol.7429,74290C:1-9,2009.
8 Fabian Duerr, Pablo Benitez.Juan C. Minano,Youri Meuretand Hugo Thienpont, Analytic design method for optimal imaging:coupling three ray sets using two free-form lens profiles, OPTICS EXPRESS Vol.20(5): 5576-5585,2012.
9 Yi Luo, Zexin Feng, Yanjun Han, Hongtao Li, Design of compact and smooth free-form optical system with uniform illuminance for LED source, OPTICS EXPRESS Vol.18(9):9055-9063,2010.
10 Rengmao Wu, Zhenrong Zheng, Haifeng Li, and Xu Liu, Optimization design of irradiance array for LED uniform rectangular illumination, Vol.51(13):2257-2263,2012.
11 Li Xu, Kexin Chen, Qingsheng He, Guofan Jin, Design of freeform mirrors in Czerny-Turner spectrometers to suppress astigmatism, Vol.(48)15:2871-2879,2009.
12 Xutao Sun, Zhenrong Zheng, Xu Liu, Peifu Gu, Etendue analysis and measurement of light source with elliptical reflector, Displays 27, pp56-61,2006.
13 Zhenrong Zheng, Design of off-axis reflective projection lens using spherical Fresnel surface, Optik 122, pp145-149,2011.
14孙旭涛,郑臻荣,刘旭,顾培夫,用Zernikc自由曲面设计弯曲屏幕超薄投影系统,浙江大学学报工学版,Vol.8,1428-1432,2009.
15程德文,王涌天,常军,刘越,徐况,轻型大视场自由曲面棱镜头盔显示器的设计,红外与激光工 程,Vo1.36(3)：309-313,2007.
16 Dewen Cheng, Yongtian Wang, Hong Hua, M. M. Talha, Design of an optical see-through head-mounted display with a low f-number and large field of view using a freeform prism, APPLIED OPTICS, Vol.48(14): 2655-2668,2009.
17 Dewen Cheng, Qingfeng Wang, Yongtian Wang, Guofan Jin, Lightweight spatial-multiplexed dual focal-plane head-mounted display using two freeform prisms, CHINESE OPTICS LETTERS Vol.11(3): 031201-1-4,2013.
18 Dewen Cheng, Yongtian Wang, Hong Hua, and Jose Sasian, Design of a wide-angle, lightweight head-mounted display using free-form optics tiling, OPTICS LETTERS Vol.36(11):2098-2100,2011.
19 Jannick P. Rolland and Kevin P. Thompson, Freeform optics:Evolution? No, revolutio! SPIE Newsroom 10.1117/2.1201207.004309, pp1-3,2012.
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