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An anamorphic imaging system, for example, consisting of cylindrical optics, is symmetric about two orthogonal planes whose intersection defines its optical axis. The Gaussian images of a point object with object rays in the two symmetry planes are formed separately. They are coincident in the final image space of the system for only two pairs of conjugate planes [1]. By definition, an anamorphic system forms the image of an extended object with different transverse magnifications in the two symmetry planes. Thus, for example, the image of a square object is rectangular and that of a rectangular object can be square. The two orthogonal planes of symmetry of the imaging system yield six “reflection” invariants in terms of the Cartesian coordinates of the object and pupil points, which become the building blocks of its aberration function for a certain point object. The six invariants reduce to three “rotational” invariants for a rotationally symmetric system, or equivalently for an infinite number of symmetry planes. In this chapter, we discuss the power series expansion of the aberration function in terms of the six reflection invariants, define the classical aberrations of the system, and discuss their balancing to minimize their variance across a rectangular exit pupil, and thereby improve the image quality [see Chapter 2]. We show that the balanced aberrations are represented by the products of the Legendre polynomials, one for each of the two dimensions of the rectangular pupil [4]. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point.
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