中文摘要：

Zernike多项式拟合是一种在光学领域中广泛应用的分析技术。由于现代光学工程中采集数据的离散性和非圆孔径系统的大量使用,Zernike多项式拟合不能完全满足分析需要。提出了一种基于Zernike多项式的非圆孔径离散采样点的正交多项式。通过矩阵的QR分解方法得到在离散采样点上的正交多项式基底。分别使用Zernike多项式和正交多项式对150 mm×90 mm的矩形光栅反射波前进行拟合,结果表明两种方法残差波前的PV和RMS值分别相差0.013波长和小于0.001波长。对比不同项数拟合的正交多项式和Zernike多项式系数表明,正交多项式系数之间彼此独立,并由正交多项式系数计算得到了对应的Seidel像差。正交多项式各项系数可以逐项求解,该方法可以显著提高求解速度。

英文摘要：

The Zernike polynomial is a widely used analytical technique in optics. Because of the discrete sampled measurement data and widely used non-circle aperture system in modern optical engineering,Zernike polynomial fitting can not satisfy a requirement completely. A kind of non-circle aperture discrete sampled orthogonal polynomial based on Zernike polynomial was proposed. The orthogonal basis was obtained using matrix QR decomposition method for discrete samples. Zernike polynomial and orthogonal polynomial were used for fitting 150 mm×90 mm rectangular grating wave-front. The differences of PV and RMS between two methods are 0.013 waves and less than 0.001 waves respectively for the residual wavefront. Comparison of different order fitting of the orthogonal polynomial and Zernike polynomial coefficients, indicate that the orthogonal polynomial coefficients are independent of each other. And the corresponding Seidel aberrations were calculated by the orthogonal polynomial coefficients. Orthogonal polynomial coefficients can be solved one by one. This method can significantly improve the solution speed.