光学自由曲面面形描述方法和光线追迹模型的研究
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摘要
随着光学系统的不断发展,光学自由曲面在特殊光学系统中的作用越发地凸显出来,但是自由曲面在面形描述和光线追迹等基础理论上的不够完善,制约了其设计、制造及测量技术的发展,限制了其广泛应用。
本文基于光学自由曲面面形描述方法和光线追迹模型的发展现状,对这两方面的基础理论进行了研究：在光学自由曲面面形描述方面,研究了以高斯函数为基函数的径向基函数面形描述法,确定基函数的个数和中心分布,通过曲面拟合求出每个基函数的系数,将基函数和其系数的乘积叠加起来得到自由曲面的面形表达式,仿真过程中将所提方法分别用于球面、非球面和扩展多项式曲面,得到了较高的拟合精度；在光线追迹模型方面,提出了基于子曲面分割和迭代的自由曲面光线追迹方法,通过离散点分割形成子曲面阵列,根据三角形内角和定理确定目标子曲面,对目标子曲面进行切平面迭代完成光线追迹,仿真过程中将所提方法分别用于偶次非球面、双圆锥曲面、扩展多项式面和径向基函数曲面,并将仿真结果与ZEMAX光线追迹结果进行比较,得到光线与曲面交点精度达到10-3nm量级,折射光线方向余弦精度达到10-3"(角秒)量级,并且具有很好的通用性和较高的光线追迹效率。
文章的第五章提出了一种轴外点消像差的自由曲面设计方法,并将上述两种理论用于所设计的自由曲面光学系统,验证了这两种理论的实用性。并且所设计的系统相对于用球面透镜设计的系统,点列图直径和波像差PV值都减小了几十倍,在成像质量上有很大提高。
With the development of optical system, free-form surface is more and more important in some special optical system. However, some related techniques are limited by the imperfect basic theory of its surface description and ray tracing method, including the designing, manufacturing and measurement.
Based on the development of free-form surface description and ray tracing method, the basic theory of free-form surface is studied in this paper. On the one hand, the radical basis functions with Gaussian form is applied to describe free-form surface shape. The number and central distribution of basis functions are determined firstly, and then the coefficients of each basis function are solved by the radical basis fitting. Finally, the shape expression of free-form surface is accumulated with the production of basis function and its coefficient. The simulation result shows that high fitting accuracy is acquired after this method applying to fit sphere, asphere and extended polynomial surface. On the other hand, a novel method based on segmentation and iteration of sub-surface is presented as the description of optical free-form surface, aiming at the complex and relatively smooth character of its figure. Three steps are executed in the whole process of ray tracing of free-form surface. Firstly, free-form surface is divided into some sub-surfaces by discrete points, and then target sub-surface is determined by angle sum of a triangle theorem. Finally, iteration algorithm is utilized on the target sub-surface. The corresponding simulation is done in MATLAB, including even asphere, biconic surface, extended polynomial surface and radial basis function surface. Compared with the tracing result in ZEMAX, the method presented is of high accuracy, good versatility and high speed. The deviation of the intersection between ray and surface is about 10-3nm, and the deviation of the direction cosine of refracted ray is about 10-3"
A novel free-form surface design method is presented to eliminate astigmatic in chapter 5. Two basis theory presented above are utilized on the system designed that verify the usefulness of those two theories. The diameter of spot diagram and PV of wavefront of free-form optical system are much less than them of the traditional optical system.
本文基于光学自由曲面面形描述方法和光线追迹模型的发展现状,对这两方面的基础理论进行了研究：在光学自由曲面面形描述方面,研究了以高斯函数为基函数的径向基函数面形描述法,确定基函数的个数和中心分布,通过曲面拟合求出每个基函数的系数,将基函数和其系数的乘积叠加起来得到自由曲面的面形表达式,仿真过程中将所提方法分别用于球面、非球面和扩展多项式曲面,得到了较高的拟合精度；在光线追迹模型方面,提出了基于子曲面分割和迭代的自由曲面光线追迹方法,通过离散点分割形成子曲面阵列,根据三角形内角和定理确定目标子曲面,对目标子曲面进行切平面迭代完成光线追迹,仿真过程中将所提方法分别用于偶次非球面、双圆锥曲面、扩展多项式面和径向基函数曲面,并将仿真结果与ZEMAX光线追迹结果进行比较,得到光线与曲面交点精度达到10-3nm量级,折射光线方向余弦精度达到10-3"(角秒)量级,并且具有很好的通用性和较高的光线追迹效率。
文章的第五章提出了一种轴外点消像差的自由曲面设计方法,并将上述两种理论用于所设计的自由曲面光学系统,验证了这两种理论的实用性。并且所设计的系统相对于用球面透镜设计的系统,点列图直径和波像差PV值都减小了几十倍,在成像质量上有很大提高。
With the development of optical system, free-form surface is more and more important in some special optical system. However, some related techniques are limited by the imperfect basic theory of its surface description and ray tracing method, including the designing, manufacturing and measurement.
Based on the development of free-form surface description and ray tracing method, the basic theory of free-form surface is studied in this paper. On the one hand, the radical basis functions with Gaussian form is applied to describe free-form surface shape. The number and central distribution of basis functions are determined firstly, and then the coefficients of each basis function are solved by the radical basis fitting. Finally, the shape expression of free-form surface is accumulated with the production of basis function and its coefficient. The simulation result shows that high fitting accuracy is acquired after this method applying to fit sphere, asphere and extended polynomial surface. On the other hand, a novel method based on segmentation and iteration of sub-surface is presented as the description of optical free-form surface, aiming at the complex and relatively smooth character of its figure. Three steps are executed in the whole process of ray tracing of free-form surface. Firstly, free-form surface is divided into some sub-surfaces by discrete points, and then target sub-surface is determined by angle sum of a triangle theorem. Finally, iteration algorithm is utilized on the target sub-surface. The corresponding simulation is done in MATLAB, including even asphere, biconic surface, extended polynomial surface and radial basis function surface. Compared with the tracing result in ZEMAX, the method presented is of high accuracy, good versatility and high speed. The deviation of the intersection between ray and surface is about 10-3nm, and the deviation of the direction cosine of refracted ray is about 10-3"
A novel free-form surface design method is presented to eliminate astigmatic in chapter 5. Two basis theory presented above are utilized on the system designed that verify the usefulness of those two theories. The diameter of spot diagram and PV of wavefront of free-form optical system are much less than them of the traditional optical system.
引文
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[2]LI sheng-yi, CHEN shan-yong, DAI Yi-fan. Inspection of Free-Form Optics. Nanotechnology and Precision Engineering,2005,3(2)
[3]李荣彬.自由曲面光学元件的设计加工及面形测量之综合制造系统.中国科技奖励,2005,5
[4]丁毅.自由曲面光学器件的设计及其在照明系统中的应用.杭州：浙江大学硕士学位论文,2009
[5]李荣彬,张志辉,杜雪,孔令豹,蒋金波.自由曲面光学元件的设计、加工及面形测量的集成制造技术.机械工程学报,2010,46(11)
[6]李荣彬,张志辉,杜雪,孔令豹,蒋金波.自由曲面光学的超精密加工技术及其应用.红外与激光工程,2010,39(1)
[7]何玉兰,刘钧,焦明印,罗传伟.利用CODE V设计含有自由曲面的光学系统.应用光学,2006,27(2)
[8]孙旭涛,郑臻荣,刘旭,顾培夫.用Zernike自由曲面设计弯曲屏幕超薄投影系统.浙江大学学报,2009,43(8)
[9]邢莎莎,吴仍茂,李海峰,郑臻荣,刘旭.应用于投影光刻离轴照明的自由曲面设计.光学学报,2011,31(3)
[10]陈云亮,李铁才,邱祥辉.头戴显示器中自由曲面棱镜的设计.应用光学,2009,30(4)
[11]Ozan Cakmakci, Jannick P.Rolland. Application of Radial Basis Functions to the Design of a Freeform Single Element See-Through Head-Worn Display. Optical Society of America,2009
[12]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.杭州：浙江大学博士学位论文,2008
[13]丁毅,顾培夫.实现均匀照明的自由曲面反射器.光学学报,2007,3
[14]王洪,张小凡,王海宏,杜乃锋.自由曲面路灯反射器设计.红外与激光工程,2010,39(4)
[15]张新,许英朝.光学自由曲面的检测方法.中国光学与应用光学,2008,1(1)
[16]何玉兰.含有自由曲面的非共轴平板显示光学系统研究.西安：西安工业大学硕士学位论文,2006
[17]K-H Brenner. Shifted Base Functions:An Efficient and Versatile New Tool in Optics. Journal of Physics,2008
[18]杨波,刘一超,王涌天.多曲面反射体的计算机辅助设计及优化.光子学报,2004,33(8)
[19]Sergio Ortiz, Damian Siedlecki, Laura Remon, Susana Marcos. Three-dimensional ray tracing on Delaunay-based reconstructed surfaces. Optical Society of America,2009, 48(20)
[20]贾丽萍,徐延海,张建武.基于Bezier曲面的子午线轮胎自由振动分析.固体力学学报,2003,24(1)
[21]李昱明,洪锡军,彭颖红.基于Bezier曲面造型的触模和摩擦条件施加算法.上海交通大学学报,2001,3(7)
[22]罗尧治,公晓莺.基于双三次B样条插值的空间结构自由曲面.空间结构,2004,10(2)
[23]周军,周太明.B样条曲面在汽车前照灯设计中的应用.照明工程学报,2000,11(4)
[24]朱心雄.自由曲线曲面造型技术.北京：科学出版社,2000
[25]佟静波,崔庆丰,薛常喜,潘春艳,张栋,张康伟.离轴双通道头盔显示器光学系统设计.光学学报,2010,30(9)
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[27]冯婕,白瑜,邢廷文Zernike多项式波面拟合精度研究.光电技术应用,2011,36(2)
[28]Rafael Navarro, Ricardo Rivera, Justiniano Aporta. Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials. Journal of Optometry,2011,4(2)
[29]Victor Arni D. P.Sicam, Joris Coppens, Tom J.T.P.vanden Berg, Rob G. L. vander Heijde. Corneal surface reconstruction algorithm that uses Zernike polynomial representation. Optical Society of America,2004,21(7)
[30]Jason Turuwhenua. Corneal surface reconstruction algorithm using Zernike polynomial representation:improvements. Optical Society of America,2007,24(6)
[31]余美群.我国人眼光学数字模型的建立.南京：南京理工大学硕士学位论文,2009
[32]关朝亮,王建敏,戴一帆,尹自强.基于Zernike多项式拟合的自由曲面车削误差补偿技术.国防科技大学学报,2010,32(2)
[33]Ozan Cakmakci, Sophie Vo, Hassan Foroosh, Jannick Rolland. Application of radial basis functions to shape description in a dual-element off-axis magnifier. Optical Society of America,2008,33(11)
[34]Ozan Cakmakcia, Ilhan Kaya, Gregory E. Fasshauerc, Kevin P. Thompsona, Jannick P. Rollandd. Application of Radial Basis Functions to Represent Optical Freeform Surfaces. Optical Society of America,2010
[35]Ozan Cakmakcia, Kevin Thompsona, Pierre Valleeb, Jasmin Coteb, Jannick P. Rollandc. Design of a Freeform Single-Element Head-Worn Display. SPIE,2010,7618
[36]K-H Brenner. General solution of two-dimensional beam-shaping with two surfaces. Journal of Physics,2010,1(206)
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