Zernike polymomials for rectangular area and applications in off-axis asphere testing

Pinjiang He; Xiaoying Zong; Ruimin Fu

doi:10.1117/12.865486

11 October 2010 Zernike polymomials for rectangular area and applications in off-axis asphere testing

Pinjiang He, Xiaoying Zong, Ruimin Fu

Proceedings Volume 7656, 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment; 765615 (2010) https://doi.org/10.1117/12.865486
Event: 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies, 2010, Dalian, China

Abstract

Zernike polynomials play a very important role in manufacture and measurement of optical element and system. They are orthogonal within the defined unit circle, which makes them having advantages in representing wave fronts and surface errors of circular regions. Zernike polynomials are not orthogonal in a rectangular or other shape region, which limits their extended applications. As optical elements with rectangular aperture are used more and more frequently in optical imaging systems such as remote sensors, there is a requirement for adapting Zernike polynomials in rectangular regions. One answer is to generate an orthogonal polynomial set by Schmidt-Grahm orhogonalization method. The expression of an orthogonal polynomial for rectangular area has been derived. A set of 2-dimension color contour figures representing the values of these polynomials has been plotted, and are compared with those of 'ordinary' Zernike Polynomials for circular region. It shows that the distribution of the 'peaks' and 'valleys' in these figures is very similar to that of 'ordinary' Zernike Polynomials. The derived Zernike Polynomials for rectangular area have been successfully applied in interferometric testing procedure of a rectangular off-axis asphere mirror. The coefficients of the polynomials in rectangular aperture are calculated and used to guide the fine alignment in the interferometric testing procedure. The tested results show that the adjustment of rectangular off-axis asphere testing system become easier and the testing efficiency is improved.

Citation Download Citation

Pinjiang He, Xiaoying Zong, and Ruimin Fu "Zernike polymomials for rectangular area and applications in off-axis asphere testing", Proc. SPIE 7656, 5th International Symposium on Advanced Optical Manufacturing and Testing Technologies: Optical Test and Measurement Technology and Equipment, 765615 (11 October 2010); https://doi.org/10.1117/12.865486

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