As discussed in the following paper, there are systems, such as the cube-corner retroreflectors, that consist of six segments in a circular pupil, with each segment in the form of a sector pupil. The orthonormal polynomials, representing the balanced aberrations for such segmented pupils, can be obtained by orthonormalizing the Zernike circle polynomials over a sector area using the Gram–Schmidt orthonormalization process.
Due to the low symmetry of the sector pupils, the closed-form analytical expressions for the polynomials are very complex; even the tilt and defocus polynomials are not simple. The complexity increases even more for a system with an annular sector pupil. In that case, there are two variables representing a point on the pupil, two parameters defining its orientation and angular subtense, and the parameter specifying its obscuration ratio. However, relatively simple expressions are obtained when the angular subtense and the orientaion of the sector pupil are specified along with its obscuration ratio.
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