Formula for the density of tangent rays over a caustic surface

Abstract

The geometrical flux density (irradiance) is singular over caustic surfaces and, therefore, cannot be used effectively as a measure of the concentration of rays at or near the caustic surfaces. A finite substitute measure, the density of rays tangent to the caustic, may be obtained by dividing an element of incident flux by the area of the caustic formed by the associated rays. This gives a measure of the energy density over different regions of the caustic. As an example, the ray density over the caustic is evaluated for collimated light reflected from a spherical mirror. A similar calculation is performed for collimated light refracted by a plano-convex singlet lens. General formulas are presented for computing the ray density over the caustic for reflection of meridional rays by an aspheric surface. Also analytical and numerical algorithms are given for evaluating the ray density over the caustic in a multiinterface optical system.