Computing Zernike polynomials of arbitrary degree using the discrete Fourier transform

Abstract

The conventional representation of Zernike polynomials Rnm(ρ) gives unacceptable numerical results for large values of the degree n. We present an algorithm for the computation of Zernike polynomials of arbitrary degree n. The algorithm has the form of a discrete Fourier (cosine) transform which comes with advantages over other methods in terms of computation time, accuracy and ease of implementation. As an application we consider the effect of NA-scaling on the lower-order aberrations of an optical system in the presence of a very high order aberration.