Analytical method for the transformation of Zernike polynomial coefficients for scaled, rotated, and translated pupils.

Zernike polynomials provide an excellent metric basis for characterizing the wavefront aberrations of human eyes and optical systems. Since the Zernike expansion is dependent on the size, position, and orientation of the pupil in which the function is defined, it is often necessary to transform the Zernike coefficients between different pupils. An analytic method of transforming the Zernike coefficients for scaled, rotated, and translated pupils is proposed in this paper. The normalized coordinate transformation functions between the polar coordinates of the transformed pupil and the Cartesian coordinates of the original pupil are given. Based on the Cartesian and polar representations of Zernike polynomials, the coefficients' transformation matrix can be derived directly and conveniently. The first 36 terms of standard Zernike polynomials are used to validate the proposed method. For different types of transformation, transformation rules of individual Zernike terms are systematically analyzed, revealing how individual terms of the original pupil transform into terms of the transformed pupil. Numerical examples are presented to demonstrate the validity of the proposed method. Further application of the proposed method to the alignment of pupil-decentered off-axis optical systems is discussed.