A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory

Abstract

A generalization of the Zernike circle polynomials for expansion of functions vanishing outside the unit disk is given. These generalized Zernike functions have the form Zm,{\alpha} n ({\rho}, \vartheta) = Rm,{\alpha} n ({\rho}) exp(im\vartheta), 0 \leq {\rho} < 1, 0 \leq \vartheta < 2{\pi}, and vanish for {\rho} > 1, where n and m are integers such that n - |m| is nonnegative and even. The radial parts are O((1 - {\rho}2){\alpha}) as {\rho} \uparrow 1 in which {\alpha} is a real parameter > -1. The Zm,{\alpha} n are orthogonal on the unit disk with respect to the weight function (1 - {\rho}2)-{\alpha}, 0 \leq {\rho} < 1. The Fourier transform of Zm,{\alpha} n can be expressed explicitly in terms of (generalized) Jinc functions Jn+{\alpha}+1(2{\pi}r)/(2{\pi}r){\alpha}+1 and exhibits a decay behaviour r-{\alpha}-3/2 as r \rightarrow \infty. Etc.