Orthonormal vector polynomials in a unit circle, Part I: Basis set derived from gradients of Zernike polynomials

Chunyu Zhao, James H. Burge

Research output: Contribution to journalArticlepeer-review

119 Scopus citations


Zernike polynomials provide a well known, orthogonal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. A related set of orthogonal functions is given here which represent vector quantities, such as mapping distortion or wavefront gradient. These functions are generated from gradients of Zernike polynomials, made orthonormal using the GramSchmidt technique. This set provides a complete basis for representing vector fields that can be defined as a gradient of some scalar function. It is then efficient to transform from the coefficients of the vector functions to the scalar Zernike polynomials that represent the function whose gradient was fit. These new vector functions have immediate application for fitting data from a Shack-Hartmann wavefront sensor or for fitting mapping distortion for optical testing. A subsequent paper gives an additional set of vector functions consisting only of rotational terms with zero divergence. The two sets together provide a complete basis that can represent all vector distributions in a circular domain.

Original languageEnglish (US)
Pages (from-to)18014-18024
Number of pages11
JournalOptics Express
Issue number26
StatePublished - Dec 24 2007

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics


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