Efficient and stable numerical method for evaluation of Zernike polynomials and their Cartesian derivatives

Abstract

Our work is focused on a problem of numerical evaluation of Zernike polynomials and their Cartesian partial derivatives. Since the direct calculation using explicit definition relations is relatively slow and it is numerically instable for higher orders of evaluated polynomials there is a need for a more effective and stable method. In recent years several recurrent methods were developed for numerical evaluation of Zernike polynomials. These methods are numerically stable up to very high orders and they are much faster than direct calculation. In our work a brief review of the existing methods for calculation of Zernike polynomials is given and then an analogous recurrence method for evaluation of x and y partial derivatives of Zernike polynomials is proposed. The numerical stability of this method and the comparison of computation time with respect to the direct method is presented using computer simulations. The proposed method can be used e.g. in optical modeling for expressing the shape of optical surfaces or in optical measurement methods based on the wavefront gradient measurements (Shack-Hartmann wavefront sensor, pyramidal sensor or shearing interferometry), where modal wavefront reconstruction using Zernike polynomials is often used.