Orthogonal systems of Zernike type in polygons and polygonal facets

TL;DR

This paper extends Zernike polynomial bases to polygonal optical apertures using a piece-wise diffeomorphism, enabling accurate wavefront representation in segmented mirror telescopes with preserved mathematical and physical properties.

Contribution

It introduces a novel method to generalize Zernike bases for polygons via a piece-wise diffeomorphism, improving over ad hoc orthonormalization approaches.

Findings

Provides explicit formulas for polygonal Zernike bases.

Ensures invariance of mathematical properties in the generalized basis.

Includes a practical example for telescope apertures.

Abstract

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Diaz et all, 2014] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piece-wise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for a typical example of telescope optical aperture are provided.