The Maréchal balancing theory for constant-amplitude circular pupils is extended to radially symmetric nonnegative amplitude distributions. A convenient technique based on the solution of a set of linear equations is proposed. It permits the determination of optimum balanced wave fronts in the weighted least-squares sense by using corresponding moments of the relevant apodizing function. Relations with the fundamental theory of orthogonal polynomials are discussed. Formulas for fast evaluation of peak intensity degradation in the far-field pattern and corresponding Maréchal tolerances are derived. Numerical results presented may be important in the design and analysis of optical systems for use with laser beams.
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