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 a hexagonal geometry, making the hexagonal orthogonal polynomials a valued basis. A cost-benefit trade-off study for deriving practical upper limits in, e.g., polishing, phasing, alignment, and stability of hexagons imposes analytical calculation to avoid time-consuming end-to-end simulations, for the sake of exactness. It is important to include global modes over the pupil for Zernike decomposition over a hexagonal segmented optical aperture 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 mode 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.
Highlights
The Zernike polynomials (e.g., [1, 2]) are commonly used in various fields of optics because they offer an orthonormal basis representing balanced classical optical aberrations defined on the unit disk over which the phase of wavefronts can be decomposed in a unique way
We present a novel approach for a rigorous Zernike and hexagonal modes decomposition adapted to hexagonal segmented pupils by means of analytical calculations
Because they are well adapted to the circular shape of most of the conventional optical systems, the Zernike polynomials became the standard way to describe optical path differences in wavefronts in many fields ranging from precision optical design and testing [3], atmospheric and adaptive optics [4,5], optical cophasing [6], vision science [7, 8], or optical communications [9]
Summary
The Zernike polynomials (e.g., [1, 2]) are commonly used in various fields of optics because they offer an orthonormal basis representing balanced classical optical aberrations defined on the unit disk over which the phase of wavefronts can be decomposed in a unique way Because they are well adapted to the circular shape of most of the conventional optical systems, the Zernike polynomials became the standard way to describe optical path differences in wavefronts in many fields ranging from precision optical design and testing [3], atmospheric and adaptive optics [4,5], optical cophasing [6], vision science [7, 8], or optical communications [9]. Control authority for the wavefront is largely assigned to a second stage of optical system and on active or adaptive optics element (i.e., small-size deformable mirrors.) In adaptive optics, advanced wavefront control devices such as small-size segmented deformable mirrors [12,13,14,15] made of hexagonal segments are intensively used in astronomy [16,17,18], laser shaping [19], retinal imaging systems [7, 20], microscopy [21, 22] or laser communication [23].
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