Conversion of Zernike polynomial coefficients to wave aberration power series coefficients

As fundamental tools in optical engineering, H. Hopkins’ wave aberration function and Zernike polynomials have respectively dominated optical design and wavefront characterization for decades. While the former underpins modern aberration analysis, the latter has become indispensable in optical testing, precision alignment, and quantitative phase evaluation at exit pupils. With the increasing complexity of contemporary optical systems requiring multidisciplinary integration, advanced simulations demand concurrent consideration of wave optical effects and geometric aberration theory, which necessitates establishing explicit connections between these two pivotal mathematical frameworks over circular apertures. Leveraging the field-dependent characteristics of Zernike polynomials, we achieve term-wise correspondence between the two polynomial systems. This approach enables rigorous derivation of an explicit transformation matrix connecting the mathematical formulations. Numerical implementations validate the proposed methodology while revealing critical insights into aberration decomposition mechanisms. The established framework provides theoretical guidance for optimizing complex optical systems, where wavefront manipulation and aberration control require coordinated treatment.