### Abstract

Rectangular pupils are employed in many optical applications such as lasers and anamorphic optics, as well as for detection and metrology systems such as some Shack-Hartmann wavefront sensors and deflectometry systems. For optical fabrication, testing, and analysis in the rectangular domain, it is important to have a well-defined set of polynomials that are orthonormal over a rectangular pupil. Since we often measure the gradient of a wavefront or surface, it is necessary to have a polynomial set that is orthogonal over a rectangular pupil in the vector domain as well. We derive curl (called C) polynomials based on two-dimensional (2-D) versions of Chebyshev polynomials of the first kind. Previous work derived a set of polynomials (called G polynomials) that are obtained from the gradients of the 2-D Chebyshev polynomials. We show how the two sets together can be used as a complete representation of any vector data in the rectangular domain. The curl polynomials themselves or the complete set of G and C polynomials has many interesting applications. Two of those applications shown are systematic error analysis and correction in deflectometry systems and mapping imaging distortion.

Original language | English (US) |
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Article number | 095105 |

Journal | Optical Engineering |

Volume | 58 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2019 |

### Keywords

- information processing
- measurement and metrology
- numerical approximation and analysis
- optical instrumentation
- surface measurements
- testing

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Engineering(all)

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## Cite this

*Optical Engineering*,

*58*(9), [095105]. https://doi.org/10.1117/1.OE.58.9.095105