光电工程, 2019, 46 (5): 180273, 网络出版: 2019-07-25
横向剪切干涉测量中一种获得无耦合Zernike系数的模式复原方法
Modal wavefront reconstruction to obtain Zernike coefficient with no cross coupling in lateral shearing measurement
Zernike圆多项式 Zernike环多项式 剪切干涉 波前复原 模式耦合 Zernike circle polynomials Zernike annular polynomials shearing interferometer wavefront reconstruc-tion modal cross coupling
摘要
模式耦合误差常见于横向剪切干涉测量中基于波前梯度数据的模式复原法,其原因是用于表示波前的基函数 ——Zernike圆多项式的导数不正交。使用一种含有 Gram矩阵的矩阵方程进行复原,直接利用 Zernike圆多项式 m≠0模式角向导数对于权重函数 w(ρ)=ρ (极坐标下)的正交性,以及 Zernike圆多项式 m=0模式径向导数对于权重函数 w(ρ)=ρ(1-ρ2)(极坐标下)的正交性进行复原。该方法无需构造辅助的向量函数,并可得到无耦合的 Zernike系数,复原结果表明,模式耦合得到了避免。该方法可推广到环上,得到无耦合的 Zernike环多项式系数。
Abstract
Modal cross coupling frequently occurs in modal approaches from wavefront gradient data such as lateral shearing measurement through Zernike circle polynomials, since the gradients of Zernike circle polynomials are not orthogonal. We use a modal approaches incorporating the Gram matrix, using the orthogonality of angular derivative of m≠0 modes with respect to weight function w(ρ)=ρ (polar coordinates), and the orthogonality of radial derivative of m=0 modes with respect to weight function w(ρ)=ρ(1-ρ2) (polar coordinates). The Gram matrix method needs no auxiliary vector functions. The Zernike coefficients can be obtained with no modal cross coupling. The simulation results are given, which indicate that the modal cross coupling is avoided by using Gram matrix method. This method can be easily extended to annulus, and the coefficients of Zernike annular polynomials with no modal cross coupling can be obtained.
孙文瀚, 王帅, 何星, 陈小君, 许冰. 横向剪切干涉测量中一种获得无耦合Zernike系数的模式复原方法[J]. 光电工程, 2019, 46(5): 180273. Sun Wenhan, Wang Shuai, He Xing, Chen Xiaojun, Xu Bing. Modal wavefront reconstruction to obtain Zernike coefficient with no cross coupling in lateral shearing measurement[J]. Opto-Electronic Engineering, 2019, 46(5): 180273.