Jacobi circle polynomials, which are orthogonal on the unit circle with orthogonal radial derivatives, have been developed previously. As the classical Zernike mode can be represented as a linear combination of Jacobi modes, Zernike wavefront modes can be reconstructed using Jacobi modes. Comparison of the Jacobi and Zernike modes for the modal approach indicates that a modal approach incorporating the Gram matrix with the Jacobi modes has potential application in high-sampling wavefront gradient sensors. The Gram matrix method using the Jacobi modes can be extended to annular pupils.
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