An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials

TL;DR

This paper introduces an automated approach to generate symbolic expressions for Zernike circular polynomials, addressing gaps in mathematical formulas and index conversion methods crucial for optics design and education.

Contribution

It presents new theorems, algorithms, and a system architecture for automatic symbolic expression generation of ZCP, improving upon existing methods.

Findings

Provides comprehensive symbolic expressions for ZCP

Offers effective algorithms for index conversion and parameter computation

Facilitates applications in optics design, software, and education

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.