3 - Path differentials

Summary

Characterizing a lens

A ray can be specified by one of its points and its direction at that point. In a homogeneous medium we can, for instance, select a ray by specifying the coordinates (x, y) of its point of intersection with the plane z = 0, and the first two components (L, M) of the unit vector in the direction of the ray. The third direction cosine N is not needed because it is the square root of (1 – L² – M²). The number of parameters needed to specify a ray is clearly four: x, y, L, and M.

Now consider a ray as it approaches a lens. It enters the lens, travels through the lens along a possibly tortuous path, and leaves the lens to enter the image space. Each of the four parameters needed to give a complete description of the ray in the image space is determined by the four parameters that specify the ray in the object space. It seems to follow that four functions of four variables are needed to characterize the lens fully: x′, y′, L′, and M′, each as a function of x, y, L, and M. In this chapter we shall see that this conclusion is incorrect; on account of Fermat's principle one single function of four variables is all that is required. As a result, Fermat's principle puts a severe constraint on the imaging processes a lens can carry out.