中文摘要：

提供了一种方形区域上归一化Zernike正交基的生成方法。它采用线性无关组Gram—Schimdt正交组构造方法，根据线性代数内积、欧氏空间及其正交性和范数的相关概念，对标准Zernike多项式进行正交处理，得到了一组新的正交多项式——Z—square多项式。采用该正交基实现了方形区域内波前模式的拟合，它不仅可由Z-square模式的集合直接对波前进行表示，而且也可以通过线性反变换，将Z—square多项式表示成标准的Zernike模式的线性组合，使被分解的波前模式与像差之间有明确的对应关系。实验表明，它不仅可以对透镜设计中的波前像差函数进行有效的拟合，而且也能对Hartmann-Shack波前传感器测试得到的实际相位数据进行拟合。

英文摘要：

An orthonormal square Zernike basis set is generated from circular Zernike polynomial apodized square mask by use of the linearly independent set Gram-Schmidt orthogonalization technique. Based on the concepts of inner product, Euclidean space and norm in the linear algebra, a standard Zernike polynomial set is made orthogonal and a new orthonormal basis of polynomials named Z-square polynomial is established. Wavefront data in square aperture can be fitted with our new orthonormal set. It can not only fit the wavefront data with Z-square basis set itself, but also can be linearly composed of standard Zernike basis set by linear reverse transform and endows the decomposed wavefront modes with a correspondent aberration meaning. The experimental results show that the Z- square polynomial set can fit the wavefront aberration data in lens design efficiently and can also fit the practical wavefront phase data of Hartmann-Shack wavefront sensor testing, it provides a method of wavefront data analysis.