Unified analytical method for Zernike coefficient transformation of scaled, rotated, and translated pupils based on Shack's vector multiplication.

Zernike polynomials play an essential role in characterizing and analyzing wavefront aberrations. Transformation of weighted coefficients for Zernike modes is required when pupil scaling, rotation, and/or translation exist. Here, a novel method based on Shack’s vector multiplication is first proposed to derive the transformation relation. The derived modes resulting from pupil scaling, rotation, and/or translation for each individual mode are easily indicated via this method; thus, the effect of each kind of pupil change could be studied qualitatively and quantitatively. Its remarkable computational efficiency against the direct integral is demonstrated by simulation. The method introduced here provides a generalized methodology to analyze the relationship between weighted coefficients for different description basis sets.