Use of Zernike polynomials for efficient estimation of orthonormal aberration coefficients over variable noncircular pupils.

An efficient way of estimating orthonormal aberration coefficients on variable noncircular pupils is proposed. The method is based on the fact that all necessary pieces of information for constructing orthonormal polynomials (via the Gram-Schmidt process) can be numerically obtained during a routine least-squares fit of Zernike polynomials to wavefront data. This allows the method to use the usual Zernike polynomial fitting with an additional procedure that swiftly estimates the desired orthonormal aberration coefficients without having to use the functional forms of orthonormal polynomials. It is also shown that the method naturally accounts for the pixelation effect of pupil geometries, intrinsic to recording wavefront data on imaging sensors (e.g., CCDs), making the coefficient estimate optimal over a given pixelated pupil geometry. With these features, the method can be ideal for real-time wavefront analysis over dynamically changing pupils, such as in the Hobby-Eberly Telescope (HET), which is otherwise inefficient with analytic methods used in past studies.