Phase retrieval on an annular pupil and an annular sector pupil by using the eigenfunction method to solve the transport of intensity equation is proposed. The analytic expressions of Laplacian eigenfunctions with the Neumann boundary condition on an annular pupil and an annular sector pupil are given. The phase can be expanded as a set of eigenfunctions on the corresponding pupil, and the coefficients of the eigenfunctions can be obtained by the integral of the eigenfunction and the intensity derivative along the optical axis. Phase retrieval by the eigen-function method on an annular pupil and an annular sector pupil is simulated, and accurate retrieved results are obtained.
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