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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 26 — Dec. 24, 2007
  • pp: 18014–18024

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

Chunyu Zhao and James H. Burge  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 18014-18024 (2007)
http://dx.doi.org/10.1364/OE.15.018014


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Abstract

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 Gram-Schmidt 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.

© 2007 Optical Society of America

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(080.1010) Geometric optics : Aberrations (global)
(220.4840) Optical design and fabrication : Testing

ToC Category:
Optical Design and Fabrication

History
Original Manuscript: October 10, 2007
Revised Manuscript: December 10, 2007
Manuscript Accepted: December 13, 2007
Published: December 18, 2007

Virtual Issues
Vol. 3, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Chunyu Zhao and James H. Burge, "Orthonormal vector polynomials in a unit circle, Part I: basis set derived from gradients of Zernike polynomials," Opt. Express 15, 18014-18024 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-26-18014


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References

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