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Orthonormal Aberration Polynomials for Anamorphic Optical Imaging Systems With Circular Pupils

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Abstract

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function [1]. The aberrations are separable in the Cartesian coordinates of a point on the pupil. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations of such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials besides being orthogonal across a rectangular pupil are inherently separable in the Cartesian coordinates of the pupil point, like the classical aberrations. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we consider an anamorphic system with a circular pupil, and show that the orthonormal aberration polynomials in this case, obtained by the Gram-Schmidt orthogonaliztion of the 2D Legendre polynomials, are not symmetrical in the two coordinates. See Table 1. For example, there is a defocus term along one axis, but no corresponding defocus term along the other axis. Similarly, there is coma or spherical aberration in one coordinate, but no corresponding term in the other coordinate. The missing defocus term, for example, is contained in another orthonormal polynomial. Is there a different way to obtain the orthogonal polynomials in question?

© 2011 Optical Society of America

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