Axial symmetry

In the previous chapters (sections 1.3, 6.6, 7.6) we have seen several cases of the image quality becoming progressively worse as the angles between the rays and the axis of the lens are increased. In this chapter we assume that the angles between the rays and the axis are so small that the images formed are essentially perfect. The resulting approximate theory of lenses is called the paraxial approximation, or Gaussian optics. We use in this chapter an abstract method based on eikonal function theory. In the next chapter we use a more down to earth approach, which links the paraxial properties of a lens to its radii, thicknesses, and refractive indices. Later on, when we deal with the problem of wave propagation through lenses, it will become clear why we need both these approaches.

The discussion will be restricted to lenses with axial symmetry around the z-axis. This restricts the possible forms of the eikonal functions, as we now demonstrate for the angle eikonal. As a first step we replace the four variables L, M, L′, and M′ by four angles. In the object space we use the slope angle ψ between the ray and the z-axis, and the azimuth angle φ between the x-axis and the projection of the ray onto the (x, y) plane. In the image space we use similar variables ψ′ and φ′.