Orthogonal polynomials: a set for square areas

Michael Bray

doi:10.1117/12.516169

26 February 2004 Orthogonal polynomials: a set for square areas

Michael Bray

Proceedings Volume 5252, Optical Fabrication, Testing, and Metrology; (2004) https://doi.org/10.1117/12.516169
Event: Optical Systems Design, 2003, St. Etienne, France

Abstract

Zernike Polynomials are classically used to analyse optical wavefronts transmitted by optical systems, most of which have circular pupils. However, square (and rectangular) areas present a problem : Zernike polynomials are not orthogonal over theses shapes, and therefore care is required in their computation. A simple way around this is to use an ... orthogonal set ! One such set is readily obtained from the 1-dimensional Legendre set of polynomials, each 2-D polynomial resulting from the product of one "x" polynomial with " y" polynomial. However, this new set has a big drawback : It lacks desirable low order terms present in the Zernike set, most notably the essential "power" term ! One answer is to generate a set of polynomials, requiring them to adhere to the "Zernike format" as far as possible. As we show in this paper, this works very well, and generates terms such as the power term mentioned above. We will present the methods mentioned above, and show the new set which is being considered for inclusion in an ISO standard on interferometry, currently being drafted by an ISO Working Group. The author calls for comments on the usefulness of the new orthogonal set presented in this paper.

Citation Download Citation

Michael Bray "Orthogonal polynomials: a set for square areas", Proc. SPIE 5252, Optical Fabrication, Testing, and Metrology, (26 February 2004); https://doi.org/10.1117/12.516169

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