Analytical decomposition of Zernike and hexagonal modes over a hexagonal segmented optical aperture

Abstract

Zernike polynomials are widely used to describe common optical aberrations of a wavefront as they are well suited to the circular geometry of various optical apertures. Non-conventional optical systems, such as future large optical telescopes with highly segmented primary mirrors or advanced wavefront control devices using segmented mirror membrane facesheets, exhibit an hexagonal geometry, making the hexagonal orthogonal polynomials a valued basis. Cost-benefit trade-off study for deriving practical upper limit in e.g., polishing, phasing, alignment, and stability of hexagons imposes analytical calculation to avoid time-consuming end-to-end simulations, and for the sake of exactness. Zernike decomposition over an hexagonal segmented optical aperture is important to include global modes over the pupil into the error budget. However, numerically calculated Zernike decomposition is not optimal due to the discontinuities at the segment boundaries that result in imperfect hexagon sampling. In this paper, we present a novel approach for a rigorous Zernike and hexagonal modes decomposition adapted to hexagonal segmented pupils by means of analytical calculations. By contrast to numerical approaches that are dependent on the sampling of the segment, the decomposition expressed analytically only relies on the number and positions of segments comprising the pupil. Our analytical method allows extremely quick results minimizing computational and memory costs. Further, the proposed formulae can be applied independently from the geometrical architecture of segmented optical apertures. Consequently, the method is universal and versatile per se. For instance, this work finds applications in optical metrology and active correction of phase aberrations. In modern astronomy with extremely large telescopes, it can contribute to sophisticated analytical specification of the contrast in the focal plane in presence of aberrations.

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