The Nature of the Paraxial Approximation II. Systems Containing Media of Radial and of Axial Non-uniform Refractive Index: Invariants

Abstract 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 (σ, τ, x y), where the coefficients are functions of the axial coordinate Z. The coefficients of the linear terms in (σ, τ; x y) are denoted by X 1(Z) and Y 1(Z), and paraxial refraction and transfer equations for the rays then appear as algebraic relations between the individual coefficients of the variables (σ, τ; x y) in X 1(Z) and Y 1(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.