### Abstract

Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

Original language | English (US) |
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Pages (from-to) | 31430-31443 |

Number of pages | 14 |

Journal | Optics Express |

Volume | 21 |

Issue number | 25 |

DOIs | |

State | Published - Dec 16 2013 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

**Orthonormal curvature polynomials over a unit circle : Basis set derived from curvatures of Zernike polynomials.** / Zhao, Chunyu; Burge, James H.

Research output: Contribution to journal › Article

*Optics Express*, vol. 21, no. 25, pp. 31430-31443. https://doi.org/10.1364/OE.21.031430

}

TY - JOUR

T1 - Orthonormal curvature polynomials over a unit circle

T2 - Basis set derived from curvatures of Zernike polynomials

AU - Zhao, Chunyu

AU - Burge, James H

PY - 2013/12/16

Y1 - 2013/12/16

N2 - Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

AB - Zernike polynomials are an orthonormal set of scalar functions over a circular domain, and are commonly used to represent wavefront phase or surface irregularity. In optical testing, slope or curvature of a surface or wavefront is sometimes measured instead, from which the surface or wavefront map is obtained. Previously we derived an orthonormal set of vector polynomials that fit to slope measurement data and yield the surface or wavefront map represented by Zernike polynomials. Here we define a 3-element curvature vector used to represent the second derivatives of a continuous surface, and derive a set of orthonormal curvature basis functions that are written in terms of Zernike polynomials. We call the new curvature functions the C polynomials. Closed form relations for the complete basis set are provided, and we show how to determine Zernike surface coefficients from the curvature data as represented by the C polynomials.

UR - http://www.scopus.com/inward/record.url?scp=84890530661&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890530661&partnerID=8YFLogxK

U2 - 10.1364/OE.21.031430

DO - 10.1364/OE.21.031430

M3 - Article

AN - SCOPUS:84890530661

VL - 21

SP - 31430

EP - 31443

JO - Optics Express

JF - Optics Express

SN - 1094-4087

IS - 25

ER -