Mathematical possibility of certain systems in geometrical optics

Abstract

A system of functional differential equations arising from Fermat’s principle is used to study qualitative questions concerning the mathematical possibility of certain rotationally symmetric piecewise homogeneous optical systems. In particular, it is shown that, given two pairs of points on the optical axis, there exist precisely two systems of single-element lenses having prescribed axial thickness and index of refraction such that these points are perfect foci. This finding sharpens an earlier result. Embedding the solution in a one-parameter family permits the construction of an asymptotic solution that requires the solution of a single nonlinear ordinary differential equation. The leading-order solution corresponds to an optical system satisfying the Herschel condition. The existence and uniqueness results are extended to optical systems having three lens boundaries such as achromatic doublets.