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

dc.contributor.authorMafusire, Cosmas
dc.contributor.authorKruger, T.P.J. (Tjaart)
dc.date.accessioned2018-10-18T10:12:40Z
dc.date.issued2018-06
dc.description.abstractThe 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.departmentPhysicsen_ZA
dc.description.embargo2019-06-01
dc.description.librarianhj2018en_ZA
dc.description.sponsorshipThe University of Pretoria (A0W679) and the National Research Foundation of South Africa (94107).en_ZA
dc.description.urihttp://www.opticsinfobase.org/josaaen_ZA
dc.identifier.citationMafusire, 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.issn1084-7529 (print)
dc.identifier.issn1520-8532 (online)
dc.identifier.other10.1364/JOSAA.35.000840
dc.identifier.urihttp://hdl.handle.net/2263/66946
dc.language.isoenen_ZA
dc.publisherOptical Society of Americaen_ZA
dc.rights© 2018 The Optical Societyen_ZA
dc.subjectOrthonormal vector circle polynomialsen_ZA
dc.subjectCartesian gradienten_ZA
dc.subjectZernike polynomialsen_ZA
dc.subjectUnit circleen_ZA
dc.subjectDerivatesen_ZA
dc.subjectLaser beamsen_ZA
dc.subjectBasis seten_ZA
dc.titleOrthonormal vector general polynomials derived from the Cartesian gradient of the orthonormal Zernike-based polynomialsen_ZA
dc.typePostprint Articleen_ZA

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