Efficient and robust recurrence relations for the Zernike circle polynomials and their derivatives in Cartesian coordinates.

Abstract

For some time it has been known and recommended that the calculation of Zernike polynomials to radial orders higher than 8 to 10 should be performed using recurrence relations rather than explicit expressions due increasingly large cancellation errors. This paper presents a set of simple recurrence relations that can be used for the unit-normalized Zernike polynomials in polar coordinates and easily adapted to Cartesian coordinates as well. The recurrence relations are also well suited for the calculation of the Cartesian derivatives of the Zernike polynomials. The recurrence relations are easily extended to arbitrarily high orders. Assessments of the precision achievable with standard 64-bit floating point arithmetic show that Zernike polynomials up to radial order 30 can be calculated over the unit disc with errors not exceeding 5E-14, and up to radial order 50 with errors not exceeding 1.2E-13. Comparison with the Zernike capability in OpticStudio (Zemax) shows that the recurrence relations are superior in performance (both speed and precision) over the existing algorithm implemented in the software. General pseudo-code for the calculation of Zernike polynomials and their derivatives is also presented.