# Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials

 dc.contributor.author Mafusire, Cosmas dc.contributor.author Kruger, T.P.J. (Tjaart) dc.date.accessioned 2018-10-18T10:12:40Z dc.date.issued 2018-06 dc.description.abstract The concept of orthonormal vector circle polynomials is revisited by deriving a set from the Cartesian gradient of Zernike polynomials in a unit circle using a matrix-based approach. The heart of this model is a closed-form matrix equation of the gradient of Zernike circle polynomials expressed as a linear combination of lower-order Zernike circle polynomials related through a gradient matrix. This is a sparse matrix whose elements are two-dimensional standard basis transverse Euclidean vectors. Using the outer product form of the Cholesky decomposition, the gradient matrix is used to calculate a new matrix, which we used to express the Cartesian gradient of the Zernike circle polynomials as a linear combination of orthonormal vector circle polynomials. Since this new matrix is singular, the orthonormal vector polynomials are recovered by reducing the matrix to its row echelon form using the Gauss–Jordan elimination method. We extend the model to derive orthonormal vector general polynomials, which are orthonormal in a general pupil by performing a similarity transformation on the gradient matrix to give its equivalent in the general pupil. The outer form of the Gram–Schmidt procedure and the Gauss–Jordan elimination method are then applied to the general pupil to generate the orthonormal vector general polynomials from the gradient of the orthonormal Zernike-based polynomials. The performance of the model is demonstrated with a simulated wavefront in a square pupil inscribed in a unit circle. en_ZA dc.description.department Physics en_ZA dc.description.embargo 2019-06-01 dc.description.librarian hj2018 en_ZA dc.description.sponsorship The University of Pretoria (A0W679) and the National Research Foundation of South Africa (94107). en_ZA dc.description.uri http://www.opticsinfobase.org/josaa en_ZA dc.identifier.citation Mafusire, C. & Kruger, T.P.J. 2018, 'Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials', Journal of the Optical Society of America A, vol. 35, no. 6, pp. 840-849. en_ZA dc.identifier.issn 1084-7529 (print) dc.identifier.issn 1520-8532 (online) dc.identifier.other 10.1364/JOSAA.35.000840 dc.identifier.uri http://hdl.handle.net/2263/66946 dc.language.iso en en_ZA dc.publisher Optical Society of America en_ZA dc.rights © 2018 The Optical Society en_ZA dc.subject Orthonormal vector circle polynomials en_ZA dc.subject Cartesian gradient en_ZA dc.subject Zernike polynomials en_ZA dc.subject Unit circle en_ZA dc.subject Derivates en_ZA dc.subject Laser beams en_ZA dc.subject Basis set en_ZA dc.title Orthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomials en_ZA dc.type Postprint Article en_ZA
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