Aspherical surface profile fitting based on the relationship between polynomial and inner products

Abstract

Abstract. High-precision aspherical polynomial fitting is essential to image quality evaluation in optical design and optimization. However, conventional fitting methods cannot reach optimal fitting precision and may somehow induce numerical ill-conditioning, such as excessively high coefficients. For this reason, a projection from polynomial equations to vector space was here proposed such that polynomial solutions could be obtained based on matrix and vector operation, so avoiding the problem of excessive coefficients. The Newton-Raphson iteration method was used to search for optimal fitting of the spherical surface. The profile fitting test showed that the proposed approach was able to obtain results with high precision and small value, which solved the numerical ill-conditioning phenomenon effectively. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Studies on digital holography for aspheric surface measurement

“…Simplified expressions or alternatives have been proposed to fit an aspheric surface, however, accuracy is the main problem here [146][147][148][149]. A projection from polynomial equations to vector space was used to avoid the ill-condition in the problem [150]. The Levenberg-Marquartdt (LM) algorithm was explored to fit aspheric surface [17] Owing to the orthogonality and least square method, the proposed method also shows good convergence speed.…”