THE NATURE OF THE PARAXIAL APPROXIMATION .2. SYSTEMS CONTAINING MEDIA OF RADIAL AND OF AXIAL NONUNIFORM REFRACTIVE-INDEX - INVARIANTS

The methods described in Part I are applied to systems containing media of non-uniform refractive index. For both radial and axial gradients, the parametric equations X = X(Z) and Y = Y(Z), and the values of the direction cosines of the ray-tangent at Z, of any ray are represented by power series in the initial object plane and entrance pupil variables (sigma, tau, x, y), where the coefficients are functions of the axial coordinate Z. The coefficients of the linear terms in (sigma, tau; x, y) are denoted by X1(Z) and Y1(Z), and paraxial refraction and transfer equations for the rays then appear as algebraic relations between the individual coefficients of the variables (sigma, tau; x, y) in X1(Z) and Y1(Z), and in the linear parts of the direction cosines of the ray-tangent at Z. Explicit forms are given, for both focusing and defocusing, radial and axial gradient-index media. It is shown that the customary optical invariant (numerical aperture x object height) is also an invariant for all of the four types of index-gradient considered. It is also shown that, for both radial and axial index-gradients, further useful invariants exist. The use of these, and of the rigorously defined paraxial approximations to the ray-paths, have facilitated the derivation of closed formulae for the primary (third-order) wavefront aberrations of systems containing media of non-uniform refractive index. These are the subject of a paper to be submitted later. The (idealised) ray-path approximations used in treating the Gaussian optics of systems are also briefly described. We now show how the methods described in Part I are applied to the different types of gradient-index media. In the interests of continuity, the section headings and diagrams are numbered in the same succession.