Analytical caustic surfaces

This document discusses the determination of caustic surfaces in terms of rays, reflectors, and wavefronts. Analytical caustics are obtained as a family of lines, a set of points, and several types of equations for geometries encountered in optics and microwave applications. Standard methods of differential geometry are applied under different approaches: directly to reflector surfaces, and alternatively, to wavefronts, to obtain analytical caustics of two sheets or branches. Gauss/Seidel aberrations are introduced into the wavefront approach, forcing the retention of all three coefficients of both the first- and the second-fundamental forms of differential geometry. An existing method for obtaining caustic surfaces through exploitation of the singularities in flux density is examined, and several constant-intensity contour maps are developed using only the intrinsic Gaussian, mean, and normal curvatures of the reflector. Numerous references are provided for extending the material of the present document to the morphologies of caustics and their associated diffraction patterns.