The Complete Basis Set of The Orthonormal Vector Polynomials in A unit Annular Circular Pupil

Abstract

In this paper, a complete set of the orthonormal vector polynomials in a unit annular pupil were derived by finding,first a set of orthogonal functions that represent vector quantities which can be generated from the gradients of annularZernike polynomials ZP, and the orthogonality is made by MATLAB code using Gram Schmidt orthogonalizationmethod. A relation of these polynomials with the circular ZP and circular ZP gradients are represented also in this work.Then, to complete the basis, a complementary set of functions were added that have zero divergence, those whichare defined locally as a rotation or curl.