  Zernike polynomials are widely used to describe the wavefront phase 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, use approximate numerical methods to reproduce a set of Zernike or hexagonal modes with the limited degree of freedom offered by hexagonal segments. 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. This work has many potential applications in particular for modern astronomy with extremely large telescopes. 
Analytical decomposition of Zernike and hexagonal modes over an   hexagonal segmented optical aperture