On the circle polynomials of Zernike and related orthogonal sets

TLDR

The paper studies polynomials in two variables that form a complete orthogonal set inside the unit circle and remain invariant under rotations about the origin. It investigates a new orthogonal set and provides generating functions for both this set and the Zernike polynomials. The authors employ a method that can derive orthogonal sets and their generating functions, applicable beyond the specific set studied. They discover that only one such set, identical to the Zernike circle polynomials, possesses simple Legendre‑like properties, enabling explicit expressions useful in phase‑contrast imaging and diffraction theory.

Abstract

ABSTRACT The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations. The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.