Exact analytical astigmatic function of aspherical surfaces

Abstract

We derive an exact analytical image astigmatic function for aspherical and hyper-aspherical surfaces of arbitrary shape that describes correctly (without any approximations) image astigmatism for the whole object space and investigate this function for two general cases: in the presence and in the absence of object astigmatism. So we discover the boundary astigmatism correction ability of aspherical surfaces. We prove that in general there are two anastigmatic points on each chief ray in every aspherical surface. We find as well analytical expressions for anastigmatic and extreme points of the function and its vertical and horizontal asymptotes. As a result we prove that in the object and image space of every refracting aspherical surface at each stop position there are two pairs of anastigmatic surfaces one of them coincides with the refracting surface itself. In the object and image space of every reflecting aspherical surface at each stop position there is in general one anastigmatic surface which coincides with the reflecting surface itself. There are special cases when the whole space of the reflecting aspherical surface is anastigmatic.