Double Zernike polynomial and its application in optical alignment

Most of the practical optical systems have circular shape lens pupil and rectangular shape image field. This paper proposes a four dimensional polynomial to describe the full-field wavefront with orthogonal polynomials set in both circular pupil and rectangular field. The basic functions of both the pupil wavefront and the field wavefront are Zernike circle polynomials, multiplying each other to a four dimensional double Zernike polynomial (DZP) function. The double Zernike polynomial coefficients of the full-field wavefront represent the global optical aberration of the image system. The misalignment perturbation changes the corresponding least squares fitted DZP coefficients. The changed DZP coefficients shows both linear and nonlinear response to the misalignment status when the misalignment is large. The Tri-Mirror Anastigmatic Telescope is used as one of the implemented example showing the changed DZP due to perturbation.