Nonrational and rational parametric descriptions of the geometric propagation of light in an optical system

Abstract

Non-rational and rational parametric interpolators are investigated and developed as mathematical entities for describing the geometric propagation of light in optical systems. The Bezier interpolator was chosen over other interpolators to describe extended and point objects and their subsequent mathematical propagation through an optical system primarily because of their superior mathematical stability, convex hull property, and endpoint interpolation, which is especially important for describing wavefront behavior. The limitations of the affine transformation first are exposed for transforming generalized three-dimensional extended Bezier objects ideally or collinearly through an optical system. This limitation necessitated the development of a projective transformation. The perspective projection next was used in a vector derivation of the collinear mapping equations thereby demonstrating that an optical collinear mapping is a special projective transformation. Furthermore, the perspective projection was found to correctly map non-rational and rational Bezier objects through an optical system. Rational Bezier interpolators, because they are inherently projections of n-dimensional functions onto hyperplanes, exist or live in a collinear space and therefore are ideally suited for describing the conjugate relationships found in optical systems. Bezier curves also are shown to describe the behavior of the meridional wavefront as it was refracted and reflected at optical surfaces. Affine maps again proved inadequate for general wavefront propagation necessitating the development of the Bezier ray trace. The Bezier ray trace was developed for both non-rational, rational quadratic, cubic, and piecewise continuous cubic orders by utilizing end control point interpolation and control polygon tangency conditions. In general non-rational quadratic and cubic Bezier curves inaccurately describe wavefront behavior in an optical system whereas their rational counterparts do so accurately. The ability of a rational Bezier curve to accurately describe a meridional wavefront leads to the interpretation that wavefronts and wave aberrations may be considered as projections of n-dimensional functions onto hyperplanes. Finally, the fourth order scalar wave aberration functions were converted into equivalent Bezier representations. This representation leads to a graphical interpretation of individual aberrations in terms of control points, which when uniformly parameterized and degree elevated to the same order, may be added by together to form composite aberrations.