A generalization of the Zernike circle polynomials for forward and inverse problems in diffraction theory
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Publication:6228300
arXiv1110.2369MaRDI QIDQ6228300
Publication date: 11 October 2011
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 ({
ho}, vartheta) = Rm,{alpha} n ({
ho}) exp(imvartheta), 0 leq {
ho} < 1, 0 leq vartheta < 2{pi}, and vanish for {
ho} > 1, where n and m are integers such that n - |m| is nonnegative and even. The radial parts are O((1 - {
ho}2){alpha}) as {
ho} 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 - {
ho}2)-{alpha}, 0 leq {
ho} < 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
ightarrow infty. Etc.
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