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

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 an analytical method for computing the matrix that enables the building of transformed Zernike coefficients from the original ones. The technique is based on the properties of Zernike polynomials and 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.