Zernike polynomials are a sequence of orthogonal polynomials that play a crucial role in optics and, in particular, modeling microscopy systems. Introduced by Frits Zernike in 1934, they are particularly useful in expressing wavefront aberrations and, thus, imperfections of imaging systems. However, their origin and properties are rarely discussed and proven. Here, we present a novel approach to Zernike polynomials using variational calculus and apply them to describe aberrations in fluorescence microscopy. In particular, we model the impact of various optical aberrations on the performance of one-photon and two-photon excitation fluorescence microscopy.

中文翻译：

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Zernike 多项式的变分微积分方法及其在 FCS 中的应用


泽尼克多项式是一系列正交多项式，在光学，特别是显微镜系统建模中发挥着至关重要的作用。它们由 Frits Zernike 于 1934 年提出，在表达波前像差以及成像系统的缺陷方面特别有用。然而，它们的起源和特性很少被讨论和证明。在这里，我们提出了一种使用变分微积分来计算 Zernike 多项式的新方法，并将其应用于描述荧光显微镜中的像差。特别是，我们模拟了各种光学像差对单光子和双光子激发荧光显微镜性能的影响。