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.
Highlights
With the rapid development of optical design and manufacturing technology, aspherical profiles have seen extensive use because of their lower rate of aberrations relative to spherical profiles.[1,2,3,4] Improving the precision of fitting projects would facilitate the evaluation of image quality in optical design and optimization significantly
While performing surface fitting with the least square method, polynomial coefficients often become far larger than maximal rise of arch of the fitted data
According to the proposed projection principle, the mutual projection between polynomials and vector could determine the relationship between the vector and the polynomial coefficients, and further determine the relationship between introduced errors and polynomial coefficient variation
Summary
With the rapid development of optical design and manufacturing technology, aspherical profiles have seen extensive use because of their lower rate of aberrations relative to spherical profiles.[1,2,3,4] Improving the precision of fitting projects would facilitate the evaluation of image quality in optical design and optimization significantly This is commonly done by increasing the number of samples or using highorder polynomials. When fitting rotational symmetric surfaces using the nonorthogonal polynomials and least square method, the peak polynomial coefficient increases with increasing polynomial order such that massive coefficients may be derived in some cases.[10] These coefficients are usually randomly oriented and several orders of magnitude higher than rise of arch, which might result in a much smaller number from the subtraction of two large numbers when calculating the rise of arch. Yang, and Hao: Aspherical surface profile fitting based on the relationship between polynomial
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