Zernike expansion is an important tool for aberration retrieval in the optical field. The Zernike coefficients in the expansion can be solved in a linear system from those focal region intensity images, which can be modeled by the extended Nijboer-Zernike approach. Here we point out that those coefficients usually follow from different prior distributions, and especially, their variances could be dramatically diverse. To incorporate the prior information, we further introduce customized penalties to those Zernike coefficients and adopt a global adaptive generative adjustment algorithm for estimating coefficients. Based on both simulated and real data, numerical experiments show that our method outperforms other conventional methods, and provides an estimate of Zernike coefficients with a low mean square error. The aberration retrieval (AR) from the intensity point-spread function in the focal region is widely used in the optical field. It usually adopts the Zernike expansion to represent the aberration linearly. The phase retrieval 1,2 , phase diversity 3,4 and curvature sensing 5,6 are three classical methods for the AR. They solve inverse problems based on the optical mechanism and statistical parameter estimation. The work 7 considered an extended Nijboer-Zernike (ENZ) diffraction from an analytic description of the focal field and realizable solutions for the aberration coefficients 7-9. A further work 10 suggested an ENZ AR method for identifying the imperfection of lens from the intensity point spread function (PSF) of the optical system. This ENZ AR method can be further applied to the high-resolution optical lithography 11. The general pupil function can be represented by a linear function of Zernike coefficients in the Zernike expansion. Through further diffraction integrals, the light field on the focal plane also has an expansion: in which the pair (r, ϕ) are polar coordinates on the image plane, the parameter f is the camera distance from the focus plane, {V m n (r, f)} n,m are contants relying on the optical system given r and f, and coefficients {β m n } are extended Nijboer-Zernike coefficients. In theory, the expansion considers infinite terms, but in practice, only the first Q terms are retained for the aberration retrieval. For example, Q = 91 was chosen to estimate those Zernike coefficients in the work 12. Moreover, it illustrates that those coefficients could follow some prior distribution and introduce the penalty mechanism into the aberration retrieval in the optical field. Here we further study the prior distribution of the Zernike coefficients by decomposing the simulated pupil function. As discussed in the work 12 , atmospheric wavefronts can be simulated using the method 13 to generate 1000 random phases. Furthermore, 1000 generalized pupil functions can be generated with those phases at constant amplitude. We consider the first 91 terms in the Zernike expansion of the generalized pupil function. Since the first term has a real coefficient, there are 181 real coefficients by separating ...
