Frequency domain analysis of surface figure fitting based on Zernike polynomials

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.