基于泽尼克多项式进行面形误差拟合的频域分析

Frequency Domain Analysis of Surface Figure Fitting Based on Zernike Polynomials

获得泽尼克多项式的频谱信息是正确利用该多项式进行误差拟合的关键。推导出了泽尼克多项式的傅里叶变换公式,在频域中分析了不同阶数该多项式的径向频谱信息和幅角频谱信息,得到了有限项泽尼克多项式能够有效表达面形误差的最大径向空间频率和角频率。基于频域分析理论,利用泽尼克多项式对不同口径局部误差进行了拟合,并利用齐戈(Zygo)干涉仪对带有不同面形误差的光学元件进行了试验分析。结果表明,当误差的径向空间频率或角频率超出泽尼克多项式所能表达的频谱范围时,拟合误差迅速变大。

Obtaining the spectrum of Zernike polynomials is the key to fit the surface errors by Zernike polynomials correctly. The Fourier transform equation of Zernike polynomials is derived to analyze both the radial spectrum and the angular spectrum information of Zernike polynomials with different orders. The maximum radial spatial frequency and angular spatial frequency of the surface errors that can be effectively generated by finite-term Zernike polynomials is obtained. Based on frequency domain theory, local errors with different apertures are fitted using Zernike polynomials and experiments on optical elements are made using Zygo interferometer. The results show that the fitting errors would increase drastically when the radial spatial frequency or the angular frequency of the error exceeds the spectrum range which could be described by Zernike polynomials.