Orthonormal polynomials in wavefront analysis: error analysis

Abstract

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions

“…He observed that it was convenient to expand functions on the disk as a Fourier series in angle combined with one-sided Jacobi polynomials in radius [107,12]. These ''Zernike polynomials'' have become very popular in optics because the lowest few terms of a Zernike expansion have a simple optical interpretation [27].…”

Orthonormal Polynomials in Wavefront Analysis: Analytical Solution

“…The error when Zernike circle polynomials are used in noncircular pupils is analyzed in a separate paper. 2 …”

Average focal length and power of a section of any defined surface

“…For example, the pupils of human eyes are generally elliptical or irregular 28 and therefore require orthonormal elliptical polynomials. 29 Dai and Mahajan 29 reported in detail the effect of the noncircular impact of the pupil on the calculation of the wavefront refractions and the significant errors introduced using Zernike polynomials for noncircular pupils, in particular for the primary common aberrations of spherical aberration, coma, and astigmatism.…”