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arXiv · 2301.01859

An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials

Hong-Yan Zhang·Yu Zhou·Fu-Yun Li
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Abstract

Zernike circular polynomials (ZCP) play a significant role in optics engineering. The symbolic expressions for ZCP are valuable for theoretic analysis and engineering designs. However, there are still two problems which remain open: firstly, there is a lack of sufficient mathematical formulas of the ZCP for optics designers; secondly the formulas for inter-conversion of Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely determinate nor satisfactory. An automatic method for generating symbolic expressions for ZCP is proposed based on five essential factors: the new theorems for converting the single/double indices of the ZCP, the robust and effective numeric algorithms for computing key parameters of ZCP, the symbolic algorithms for generating mathematical expressions of ZCP, and meta-programming \& \LaTeX{} programming for generating the table of ZCP. The theorems, method, algorithms and system architecture proposed are beneficial to both optics design process, optics software, computer-output typesetting in publishing industry as well as STEM education.

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Hong-Yan Zhang, Yu Zhou, Fu-Yun Li. 2023-01-05. An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials. https://doi.org/10.1109/access.2023.3283028

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