Characteristic Functions

Abstract

In Chapter 1 we developed geometrical optics on the basis of Fermat’s principle. In this chapter, we introduce certain characteristic functions, first studied by Hamilton, that characterize the properties of the system completely. The usefulness of this method lies in the fact that just by knowing certain symmetries possessed by the system, one can draw general conclusions about the performance of the system. For example, it will be shown that the third-order aberrations of any rotationally symmetric system, whether it is made up of homogeneous media separated by surfaces or has a continuous variation of refractive index, can be completely specified by five aberration coefficients, which are known as the Seidel aberrations. It will also be shown that using the characteristic functions it is possible to obtain the characteristics of optical systems with desired properties. For example, one can determine a surface that focuses all rays emanating from a single point onto another point (see Problem 3.1) or one can determine the equation of a surface that renders parallel all rays emanating from a single point, after a reflection (see Problem 3.5). At the same time it should be pointed out that it is, in general, difficult to calculate the characteristic functions for a given system exactly, although even approximate forms give good results. In fact, the lowest-order approximation yields the paraxial properties of the system, and higher and higher orders of approximations give the various orders of aberrations present in the system.