Variational calculus approach to Zernike polynomials with application to FCS

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.