Numerically stable iterative method for the inversion of wave-front aberrations from measured point-spread-function data

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Abstract
This paper outlines a method for the determination of the unknown wave-front aberration function of an optical system from noisy measurements of the corresponding point-spread function. The problem is cast as a nonlinear least-squares estimation problem for the values of the wave-front aberration function at N points over the slit aperture, from measurements of the point-spread function at M points with M≥N. Newton’s method is used to replace the nonlinear minimization problem with a sequence of linear problems. Each such problem requires the inversion of the Hessian matrix of the error metric that is shown to be both singular (with rank ≤N - 1) and ill-conditioned. To overcome singularity, the pseudoinverse is used; to overcome ill-conditioning the pseudoinverse is calculated using singular value decomposition and the singular values then filtered. Attention is drawn to difficulties such as nonuniqueness, sensitivity of algorithms to initial guess, etc.; the ancillary mathematical details being set out in appendices. Some illustrative numerical results are presented and analyzed.

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