Vector polynomials for direct analysis of circular wavefront slope data

Abstract

In the aberration analysis of a circular wavefront, Zernike circle polynomials are used to obtain its wave aberration coefficients. To obtain these coefficients from the wavefront slope data, we need vector functions that are orthogonal to the gradients of the Zernike polynomials, and are irrotational so as to propagate minimum uncorrelated random noise from the data to the coefficients. In this paper, we derive such vector functions, which happen to be polynomials.

Wavefront analysis from its slope data

“…However, only 12 of the first 45 vector functions for which n £ 8 don't belong to the m = 0, m n = , or m n = -2 category. They are for n m , ( ) = (5, 1), (6, 2), (7, 1), (7,3), (8,2) and (8,4). As expected, the annular vector functions reduce to the corresponding vector polynomials for a circular wavefront as ⑀ AE 0.…”

“…Accordingly, we obtain them in Section 5 as the gradient of a scalar function. We find that these functions are polynomials for a circular wavefront [7], but general functions for an annular wavefront, as discussed very briefly in Section 6. [8].…”

“…The vector functions obtained as the gradient of a scalar function are irrotational. The vector functions for a circular wavefront are polynomials [7]. Both the Zernike and vector polynomials are given in Tables 1 and 2 for primary aberrations.…”

“…Is there a systematic way to choose the right set of vector functions? Let us consider the effect of noise in the slope data on the Zernike coefficients We show that when the data are noisy, which inevitably they are, irrotational vector functions (i.e., those whose curls are zero) [5][6][7] propagate the least noise from the data to the coefficients. This derivation is independent of the shape of the pupil, e.g., it is applicable to annular pupils.…”

“…We outline the differences of the annular vector functions [8] from the vector polynomials for a circular pupil [7]. The radial integrations for an annular pupil are from ⑀ to 1, where ⑀ is its obscuration ratio (instead of from 0 to 1, as for a circular pupil), and we use the annular polynomials [1] in Eq.…”