The characteristic functions of Hamilton

IN §3.1 it was shown that, within the approximations of geometrical optics, the field may be characterized by a single scalar function S(r). Since S(r) satisfies the eikonal equation §3.1 (15), this function is fully specified by the refractive index function (r) alone, together with the appropriate boundary conditions.

Instead of the function S(r), closely related functions known as characteristic functions of the medium are often used. They were introduced into optics by W. R. Hamilton, in a series of classical papers. Although on account of algebraic complexity it is impossible to determine the characteristic functions explicitly for all but the simplest media, Hamilton's methods nevertheless form a very powerful tool for systematic analytical investigations of the general properties of optical systems.

In discussing the properties of these functions and their applications, an isotropic but generally heterogeneous medium will be assumed.

The point characteristic

Let (x0, y0, z0) and (x1, y1, z1) be respectively the coordinates of two points PQ and P\ each referred to a different set of mutually parallel, rectangular axes (Fig. 4.1). If the two points are imagined to be joined by all possible curves, there will, in general, be some amongst them, the optical rays, which satisfy Fermat's principle. Assume for the present that not more than one ray joins any two arbitrary points.