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Optimum balanced wave-front aberrations for radially symmetric amplitude distributions: Generalizations of Zernike polynomials

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Abstract

The Zernike aberration theory for constant amplitude circular apertures is extended to annular apertures having a Gaussian-like radial taper. Explicit expressions are obtained for the optimum balanced wave-front aberrations in terms of shifted Jacobi polynomials. Properties of the polynomials (e.g., Rodrigues formula, recurrence relations, derivatives, etc.) are investigated.

© 1980 Optical Society of America

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