Zernike functions are orthogonal within the unit circle, but they are not over the discrete points such as CCD arrays or finite element grids. This will result in reconstruction errors for loss of orthogonality. By using roots of Legendre polynomials, a set of points within the unit circle can be constructed so that Zernike functions over the set are discretely orthogonal. Besides that, the location tolerances of the points are studied by perturbation analysis, and the requirements of the positioning precision are not very strict. Computer simulations show that this approach provides a very accurate wavefront reconstruction with the proposed sampling set.
© 2012 Optical Society of America
Original Manuscript: March 20, 2012
Revised Manuscript: April 28, 2012
Manuscript Accepted: April 29, 2012
Published: June 18, 2012
Zhenguang Shi, Yongxin Sui, Zhenyu Liu, Ji Peng, and Huaijiang Yang, "Mathematical construction and perturbation analysis of Zernike discrete orthogonal points," Appl. Opt. 51, 4210-4214 (2012)