2. Key Laboratory on Adaptive Optics, Chinese Academy of Sciences, Chengdu, Sichuan 610209, China;
3. University of the Chinese Academy of Sciences, Beijing 100049, China
Overview: Modal cross coupling frequently occurs in modal approaches from wavefront gradient data such as lateral shearing measurement through Zernike circle polynomials, since the gradients of Zernike circle polynomials are not orthogonal. We use a modal approach incorporating the Gram matrix, instead of least squares estimation, to reconstruct coefficients of modes for high sampling gradient measurement such as lateral shearing measurement. The matrix equation incorporating the Gram matrix has exactly one solution when the modes of the Gram matrix are linearly dependent. The matrix equation incorporating the Gram matrix has the solution without modal cross coupling when the modes of the Gram matrix are mutually orthogonal with respect to the same weight function of the Gram matrix. Using the orthogonality of angular derivative of m≠0 modes with respect to weight function w(ρ) = ρ (polar coordinates), one can obtain Zernike coefficients of m≠0 modes without modal cross coupling by Gram matrix method. Using the orthogonality of radial derivative of m = 0 modes with respect to weight function w(ρ) = ρ(1-ρ2) (polar coordinates), one can obtain Zernike coefficients of m≠0 modes without modal cross coupling by Gram matrix method. The Gram matrix method needs no auxiliary vector functions, and can be easily constructed and calculated. The Zernike coefficients can be obtained with no modal cross coupling. The numerical simulation results are given. Remaining error can characterize the modal cross coupling when sampling number is sufficiently high so that modal aliasing is able to be neglected. The numerical simulation result shows that the remaining error keeps very small as the truncation number J changes. The result indicates that the modal cross coupling is avoided by using Gram matrix method. This method can be easily generalized to annulus, one can obtain Zernike annular polynomial coefficients with no modal cross coupling.