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

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