Vector functions for direct analysis of annular wavefront slope data

Abstract

In the aberration analysis of a wavefront over a certain domain, the polynomials that are orthogonal over and represent balanced aberrations for this domain are used. For example, Zernike circle polynomials are used for analysis of a circular wavefront. When the data available for analysis are the slopes of a wavefront, the polynomials used are those that are orthogonal to the gradient of Zernike polynomials, and are irrotational. Similarly, the annular polynomials are used to analyze the annular wavefronts for systems with annular pupils, as in a rotationally symmetric two-mirror system. In this paper we derive the vector functions that are orthogonal to the gradients of annular polynomials and are irrotational so that they propagate minimum noise from the slope data to the annular aberration coefficients. These vector functions can be used directly to obtain independent annular aberration coefficients as their inner products with the slope data of annular wavefronts. The utility of the functions is demonstrated in a numerical simulation of noisy wavefront slope data to determine the annular coefficients.