透射波前Zernike系数与波长的函数关系研究
|
摘要
波前像差可以很好地反映光学系统的性能,波前检测结果通常用一组Zernike多项式线性组合表示。透射式光学系统波前需要在特定波长下检测,由于检测仪器的限制,目前只有少数波长的波前可以得到准确检测。提出一种新的思路,通过分析透射波前Zernike系数与波长的函数关系,间接反映波前随波长的变化规律,从而实现任意波长波前的检测。利用Zemax对光学系统建模,采集光学系统在不同波长时的波前Zernike系数,通过Matlab曲线拟合工具寻找Zernike系数与波长曲线可能存在的函数形式,并最终确定了Conrady-Zernike公式。模拟的光学系统使用这种解析式计算所得Zernike系数的最大误差在1%以内。结果表明透射式光学系统的波前Zernike系数与波长之间基本符合Conrady-Zernike公式的关系。
Wavefront aberration can well reflect the performances of optical system.The test result of wavefront is usually expressed in a linear combination of Zernike polynomials.The wavefront of the transmission optical system needs to be tested at specific wavelengths.From now on,only a few wavelength wavefronts can be accurately tested because of the limitation of detection instrument.A new idea is put forward in this paper,the wavefront can be tested at any wavelength by analyzing the functional relationship between transmitted wavefront Zernike coefficients and wavelength,which can indirectly reflect the variation of the wavefront with the wavelength.The optical system is modeled by Zemax.The wavefront Zernike coefficients of the optical system at different wavelengths are collected.Then the possible function form of the curve of the Zernike coefficients and the wavelength are fitted by Matlab curve fitting tool.Finally,the Conrady-Zernike formula is determined.The maximum error of the calculated Zernike coefficients is within 1%,which is calculated by the Conrady-Zernike formula in the simulated optical system.The results show that the wavefront Zernike coefficients and the wavelength of transmitted optical system are basically consistent with the Conrady-Zernike formula.
Wavefront aberration can well reflect the performances of optical system.The test result of wavefront is usually expressed in a linear combination of Zernike polynomials.The wavefront of the transmission optical system needs to be tested at specific wavelengths.From now on,only a few wavelength wavefronts can be accurately tested because of the limitation of detection instrument.A new idea is put forward in this paper,the wavefront can be tested at any wavelength by analyzing the functional relationship between transmitted wavefront Zernike coefficients and wavelength,which can indirectly reflect the variation of the wavefront with the wavelength.The optical system is modeled by Zemax.The wavefront Zernike coefficients of the optical system at different wavelengths are collected.Then the possible function form of the curve of the Zernike coefficients and the wavelength are fitted by Matlab curve fitting tool.Finally,the Conrady-Zernike formula is determined.The maximum error of the calculated Zernike coefficients is within 1%,which is calculated by the Conrady-Zernike formula in the simulated optical system.The results show that the wavefront Zernike coefficients and the wavelength of transmitted optical system are basically consistent with the Conrady-Zernike formula.
引文
[1]Shan B Z,Wang S Y,Niu H B,et al.Zernike polynomial fitting method and its application[J].Optics and Precision Engineering,2002,10(3):318-323.单宝忠,王淑岩,牛憨笨,等.Zernike多项式拟合方法及应用[J].光学精密工程,2002,10(3):318-323.
[2]Shao J,Ma D M,Nie Z W.Aberration analysis for annular pupils by different Zernike polynomials[J].Journal of Applied Optics,2010,31(4):544-548.邵晶,马冬梅,聂真威.不同Zernike多项式求取环孔径波面像差的研究[J].应用光学,2010,31(4):544-548.
[3]Hou X,Wu F,Yang L,et al.Effect of central obscuration interferograms fitted with Zernike circle polynomials on calculating Seidel aberrations[J].Acta Optica Sinica,2006,26(1):54-60.侯溪,伍凡,杨力,等.中心遮拦干涉图的圆泽尼克拟合对计算赛德尔像差的影响分析[J].光学学报,2006,26(1):54-60.
[4]Huang Y.Study on optimization of full field aberration of projection lens based on the extended Zernike polynomials[D].Beijing:University of Chinese Academy of Sciences,2015:8-22.黄杨.基于扩展Zernike多项式的投影物镜全视场像差优化研究[D].北京:中国科学院大学,2015:8-22.
[5]Wyant J C,Creath K.Basic wavefront aberration theory for optical metrology[C].Applied Optics&Optical Engineering,1992,XI:2.
[6]Chen L X,Hu X C,Han K,et al.Improvement method for fitting high-frequency phase by Zernike polynomials[J].Acta Optica Sinica,2016,36(3):0314001.陈丽霞,胡小川,韩开,等.Zernike多项式对空间高频相位拟合的改进方法[J].光学学报,2016,36(3):0314001.
[7]Yu X.Research on technology of computer aided alignment of off-axis four-mirror anastigmat optical system[D].Beijing:University of Chinese Academy of Sciences,2015:34-39.于鑫.离轴四反系统的计算机辅助装调技术研究[D].北京:中国科学院大学,2015:34-39.
[8]Gong D,Tian T Y,Wang H.Computer-aided alignment of off-axis three-mirror system by using Zernike coefficients[J].Optics and Precision Engineering,2010,18(8):1754-1759.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.
[9]Mahajan V N.Zernike polynomials and wavefront fitting[M]∥John Wiley&Sons,2006:498-546.
[10]Feng J,Bai Y,Xing Y W.Fitting accuracy of wavefront using Zernike polynomials[J].ElectroOptic Technology Application,2011,26(2):31-34.冯婕,白瑜,邢廷文.Zernike多项式波面拟合精度研究[J].光电技术应用,2011,26(2):31-34.
[11]Chen L F,Yang J Y,Cui S,et al.Matlab simulation of curve fitting based on least-squares[J].Journal of Shenyang Normal University(Natural Science Edition),2014,32(1):75-79.陈岚峰,杨静瑜,崔崧,等.基于Matlab的最小二乘曲线拟合仿真研究[J].沈阳师范大学学报(自然科学版),2014,32(1):75-79.
[12]Song X X.A general data fitting method based on Matlab[J].Journal of Shanxi Datong University(Natural Science),2014,30(4):1-3.宋晓霞.基于Matlab的通用数据拟合方法[J].山西大同大学学报(自然科学版),2014,30(4):1-3.
[13]Wang W S,Liu D M,Xiang Y,et al.Applied optics[M].Wuhan:Huazhong University of Science and Technology Press,2010:167-177.王文生,刘冬梅,向阳,等.应用光学[M].武汉:华中科技大学出版社,2010:167-177.
[14]Han S,Joseph A,Artur G,et al.Design of an interferometer for the measurement of long radius optics[C].SPIE,2000,3966:426-435.
[2]Shao J,Ma D M,Nie Z W.Aberration analysis for annular pupils by different Zernike polynomials[J].Journal of Applied Optics,2010,31(4):544-548.邵晶,马冬梅,聂真威.不同Zernike多项式求取环孔径波面像差的研究[J].应用光学,2010,31(4):544-548.
[3]Hou X,Wu F,Yang L,et al.Effect of central obscuration interferograms fitted with Zernike circle polynomials on calculating Seidel aberrations[J].Acta Optica Sinica,2006,26(1):54-60.侯溪,伍凡,杨力,等.中心遮拦干涉图的圆泽尼克拟合对计算赛德尔像差的影响分析[J].光学学报,2006,26(1):54-60.
[4]Huang Y.Study on optimization of full field aberration of projection lens based on the extended Zernike polynomials[D].Beijing:University of Chinese Academy of Sciences,2015:8-22.黄杨.基于扩展Zernike多项式的投影物镜全视场像差优化研究[D].北京:中国科学院大学,2015:8-22.
[5]Wyant J C,Creath K.Basic wavefront aberration theory for optical metrology[C].Applied Optics&Optical Engineering,1992,XI:2.
[6]Chen L X,Hu X C,Han K,et al.Improvement method for fitting high-frequency phase by Zernike polynomials[J].Acta Optica Sinica,2016,36(3):0314001.陈丽霞,胡小川,韩开,等.Zernike多项式对空间高频相位拟合的改进方法[J].光学学报,2016,36(3):0314001.
[7]Yu X.Research on technology of computer aided alignment of off-axis four-mirror anastigmat optical system[D].Beijing:University of Chinese Academy of Sciences,2015:34-39.于鑫.离轴四反系统的计算机辅助装调技术研究[D].北京:中国科学院大学,2015:34-39.
[8]Gong D,Tian T Y,Wang H.Computer-aided alignment of off-axis three-mirror system by using Zernike coefficients[J].Optics and Precision Engineering,2010,18(8):1754-1759.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.
[9]Mahajan V N.Zernike polynomials and wavefront fitting[M]∥John Wiley&Sons,2006:498-546.
[10]Feng J,Bai Y,Xing Y W.Fitting accuracy of wavefront using Zernike polynomials[J].ElectroOptic Technology Application,2011,26(2):31-34.冯婕,白瑜,邢廷文.Zernike多项式波面拟合精度研究[J].光电技术应用,2011,26(2):31-34.
[11]Chen L F,Yang J Y,Cui S,et al.Matlab simulation of curve fitting based on least-squares[J].Journal of Shenyang Normal University(Natural Science Edition),2014,32(1):75-79.陈岚峰,杨静瑜,崔崧,等.基于Matlab的最小二乘曲线拟合仿真研究[J].沈阳师范大学学报(自然科学版),2014,32(1):75-79.
[12]Song X X.A general data fitting method based on Matlab[J].Journal of Shanxi Datong University(Natural Science),2014,30(4):1-3.宋晓霞.基于Matlab的通用数据拟合方法[J].山西大同大学学报(自然科学版),2014,30(4):1-3.
[13]Wang W S,Liu D M,Xiang Y,et al.Applied optics[M].Wuhan:Huazhong University of Science and Technology Press,2010:167-177.王文生,刘冬梅,向阳,等.应用光学[M].武汉:华中科技大学出版社,2010:167-177.
[14]Han S,Joseph A,Artur G,et al.Design of an interferometer for the measurement of long radius optics[C].SPIE,2000,3966:426-435.