## 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. These aberrations are inherently separable in the Cartesian coordinates $(x,y)$ of a point on the pupil. Accordingly, there is $x$-defocus and $x$-coma, $y$-defocus and $y$-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, ${L}_{l}(x){L}_{m}(y)$, where $l$ and $m$ are positive integers (including zero) and ${L}_{l}(x)$, for example, represents an orthonormal Legendre polynomial of degree $l$ in $x$. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial ${L}_{l}(x){L}_{m}(y)$, there is a corresponding orthonormal polynomial ${L}_{l}(y){L}_{m}(x)$ obtained by interchanging $x$ and $y$. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram–Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in $x$ and $y$, there is no corresponding polynomial obtained by interchanging $x$ and $y$. For example, there are polynomials representing $x$-defocus, balanced $x$-coma, and balanced $x$-spherical aberration, but no corresponding $y$-aberration polynomials. The missing $y$-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.

©2012 Optical Society of America

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