Variational calculus approach to Zernike polynomials with application to fluorescence correlation spectroscopy

Zernike polynomials are a sequence of orthogonal polynomials that play a crucial role in optics and in particular in modeling microscopy systems. First 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 the description of aberrations in fluorescence microscopy. In particular, we model the impact of various optical aberrations on the performance of one-photon and two-photon excitation microscopy. SIGNIFICANCE This manuscript explores the mathematical derivation of Zernike polynomials and highlights their critical role in describing optical wavefronts and aberrations, particularly in the domain of optical microscopy. Special emphasis is placed on their utility in simulating the effects of aberrations on one-photon and two-photon excitation fluorescence correlation spectroscopy. ### Competing Interest Statement The authors have declared no competing interest.