Zernike polynomials play an essential role in characterizing and analyzing wavefront aberrations. Transformation of weighted coefficients for Zernike modes is required when pupil scaling, rotation, and/or translation exist. Here, a novel method based on Shack’s vector multiplication is first proposed to derive the transformation relation. The derived modes resulting from pupil scaling, rotation, and/or translation for each individual mode are easily indicated via this method; thus, the effect of each kind of pupil change could be studied qualitatively and quantitatively. Its remarkable computational efficiency against the direct integral is demonstrated by simulation. The method introduced here provides a generalized methodology to analyze the relationship between weighted coefficients for different description basis sets.

中文翻译：

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基于Shack向量乘法的缩放、旋转和平移瞳孔Zernike系数变换的统一解析方法

Zernike 多项式在表征和分析波前像差方面发挥着重要作用。当存在瞳孔缩放、旋转和/或平移时，需要对 Zernike 模式的加权系数进行转换。在这里，首先提出了一种基于Shack向量乘法的新方法来推导变换关系。通过这种方法可以很容易地指示每个单独模式的由瞳孔缩放、旋转和/或平移产生的派生模式；因此，可以定性和定量地研究每种瞳孔变化的影响。仿真证明了其对直接积分的显着计算效率。这里介绍的方法提供了一种通用的方法来分析不同描述基组的加权系数之间的关系。