A Different Approach to Lighting and Imaging: Formulas for Flux Density, Exact Lens and Mirror Equations and Caustic Surfaces in Terms of the Differential Geometry of Surfaces

Abstract

A formula is derived for the flux density associated with each ray traced through an optical system. The formula involves the ratio of the products of the principal curvatures of the wave front as it approaches and leaves each refracting surface. As input to the flux equation, a new and simplified derivation of the general lens equations is given. The general lens equations yield the normal curvatures and torsion of normal curves in the refracted wave front at each surface; these in turn, are related to the corresponding quantities for the incident wave front and the refracting surface. As an application, the flux equation and the generalized lens equations are specialized to meridional rays. The flux density and the caustic surfaces, that is, the loci of wave front principal curvatures, are then computed for a singlet lens. In a second application the flux density for skew rays is calculated over a receiver plane perpendicular to the symmetry axis when light from an off axis point source is reflected from a paraboloid.