Differential method for determining third-order derivative matrix of ray with respect to source ray variables at spherical boundary

The two most common optical boundaries in geometrical optics are the spherical and flat. The present group recently derived the third-order derivative matrix of a skew ray with respect to the source ray vector for a ray reflected/refracted at a flat boundary. The proposed method was based on a differential geometry approach, and hence had the advantages of an improved accuracy and the need to trace just one ray. In the present study, the method is extended to a ray incident on a spherical boundary. The derived matrix is used to explore the primary wavefront spherical aberration of an axis-symmetrical system. Its result is identical to that obtained from Zemax simulations. The estimation capability of the fifth-order Taylor series expansion of a ray is investigated by using the finite difference methods and the developed matrix. The proposed matrix can serve as a useful basis for determining the higher-order differential derivative matrices of a ray to explore higher-order ray or wavefront aberrations in future studies.