Fourier-Mellin expansion coefficients of scaled pupils

Orthogonal polynomials over the interior of a unit circle are widely used in aberration theory and in describing ocular wavefront in ophthalmic applications. In optics, Zernike polynomials (ZPs) are commonly applied for the same purpose, and scaling their expansion coefficients to arbitrary aperture sizes is a useful technique to analyze systems with different pupil sizes. By employing the orthogonal Fourier-Mellin polynomials and their properties, a new formula is established based on the same techniques used to develop the scaled pupil sizes. The description by the orthogonal Fourier-Mellin polynomials for the aberration functions is better than that by the ZPs in terms of the wavefront reconstruction errors.