An Automatic Method for Generating Symbolic Expressions of Zernike Circular Polynomials

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 remains open: firstly, there is a lack of sufficient mathematical formula of the ZCP for optics designers; secondly the formula for inter-conversion of Noll's single index and Born-Wolf's double indices of ZCP are neither uniquely detertermined 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 optics design process and software.