Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials.
Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials.
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DOI：
10.1364/josaa.30.002539
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发表时间：
2013-12
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影响因子：
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通讯作者：
Wenqing Sun;Lei Chen;Wulan Tuya;Yong He;R. Zhu
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文献类型：
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作者：
Wenqing Sun;Lei Chen;Wulan Tuya;Yong He;R. Zhu
切比雪夫和勒让德多项式经常用于矩形光瞳中的波前近似。理想情况下，数据集完全符合多项式基础，提供全光瞳近似系数和相应的几何像差。然而，如果存在水平平移和缩放，则原始多项式中的各项将变成其他项的系数的线性组合。本文介绍了平移和缩放后两种典型情况的解析表达式。通过小的平移，可以使用一阶泰勒展开来简化计算。可以选择几个代表性项作为输入来计算平移和缩放之前和之后的系数变化。结果表明，解析解的结果与离散抽样下的近似值是一致的。通过计算一组随机生成的系数，我们对比了不同平移和缩放条件下的变化。比率越大，近似值与原始值的偏差越大。最后，我们分析了使用一阶近似和直接展开在不同平移值下的峰谷（PV）和均方根（RMS）偏差。结果表明，当平移小于4%时，一阶1D-Legendre展开中偏差最大的第5项PV偏差小于7%，RMS偏差小于2%。本文给出的矩形区域平移和缩放过程中多重典型函数基的解析表达式和离散采样下的计算结果可应用于波前逼近和分析。
Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.