Transformation of Zernike coefficients: A Fourier based method for scaled, translated and rotated wavefront apertures

TL;DR

This paper introduces a Fourier-based analytical method to accurately compute how Zernike coefficients transform under aperture scaling, translation, and rotation, facilitating improved wavefront analysis in optical systems.

Contribution

It presents a novel analytical approach using Fourier properties of Zernike polynomials to derive exact transformation matrices for aperture modifications.

Findings

Exact integral formulas for transformation matrices

Applicable to full apertures without central obstruction

Does not depend on polynomial ordering

Abstract

This paper studies the effects on Zernike coefficients of aperture scaling, translation and rotation, when a given aberrated wavefront is described on the Zernike polynomial basis. It proposes a new analytical method for computing the matrix that enables the building of the transformed Zernike coefficients from the original ones. The technique is based on the properties of Zernike polynomials Fourier Transform and, in the case of a full aperture without central obstruction, the coefficients of the matrix are given in terms of integrals of Bessel functions. The integral formulas are exact and do not depend on any specific ordering of the polynomials.