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. The matrix equation incorporating the Gram matrix has exactly one solution when the modes of the Gram matrix are linearly dependent. 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. 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. Zernike annular polynomials and optical aberrations of systems with annular pupils. Cambridge: Cambridge University Press, 1999: 94. Singapore: World Scientific, 1989: 139,169–173.Īndrews G E, Askey R, Roy R. Cambridge: Cambridge University Press, 1999: 905–910. Monthly Notices of the Royal Astronomical Society, 2002, 94(2): 377–384. Diffraction theory of the knife-edge test and its improved form, the phase-contrast method. Zernike circle polynomials and optical aberrations of systems with circular pupils. Cambridge: Cambridge University Press, 1990: 407. Jacobi circle and annular polynomials: modal wavefront reconstruction from wavefront gradient. Vector polynomials for direct analysis of circular wavefront slope data. Orthonormal vector polynomials in a unit circle, Part Ⅱ: completing the basis set. Orthonormal vector polynomials in a unit circle, Part Ⅰ: basis set derived from gradients of Zernike polynomials. Vector polynomials orthogonal to the gradient of Zernike polynomials. Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian. Journal of the Optical Society of America A, 2012, 29(4): 513–520. Phase retrieval on annular and annular sector pupils by using the eigenfunction method to solve the transport of intensity equation. Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing. Journal of the Optical Society of America, 1981, 71(8): 989–992. Cross coupling and aliasing in modal wave-front estimation. Journal of the Optical Society of America, 1979, 69(7): 972–977. Modal wave-front estimation from phase derivative measurements. Wavefront phase recovery from the plenoptic camera. Measurement of the wavefront phase of a laser beam with Hartmann-Shack sensor. Reconstruction of turbulent optical wavefront realized by Zernike polynomial. Journal of the Optical Society of America, 1979, 69(3): 393–399. Matrix formulation of the reconstruction of phase values from phase differences. Zernike polynomial fitting of lateral shearing interferometry. Analysis of lateral shearing interferograms by use of Zernike polynomials. Method for evaluating lateral shearing interferograms. This method can be easily extended to annulus, and the coefficients of Zernike annular polynomials with no modal cross coupling can be obtained. The simulation results are given, which indicate that the modal cross coupling is avoided by using Gram matrix method. The Zernike coefficients can be obtained with no modal cross coupling. The Gram matrix method needs no auxiliary vector functions. We use a modal approaches incorporating the Gram matrix, using the orthogonality of angular derivative of m≠0 modes with respect to weight function w( ρ) = ρ (polar coordinates), and the orthogonality of radial derivative of m = 0 modes with respect to weight function w( ρ) = ρ(1- ρ 2) (polar coordinates).
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.
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