Prentice's equation and generalizations

Abstract

The purpose of this note is to clarify confusion over the range of applicability of Prentice's equation for the calculation of prismatic effects in lenses. Making use of the concept of the ray transference in Gaussian and linear optics, the paper obtains generalized forms of Prentice's equation that apply in various situations. Redefinition of prismatic deviation as change in reduced direction of a ray makes Prentice's equation valid for thin systems in media of any index of refraction. In particular, the equation then holds for a single refracting surface. It does not hold in general for thick systems, although there are some special cases in which it does apply. The appropriate equation is presented for thick systems in general. The correct form of power to use in the equations is equivalent power and not back-vertex power. A generalized form of Prentice's equation applies to astigmatic systems, including single refracting surfaces and thin lenses. An analog of Prentice's equation gives the transverse displacement of a ray across a system as a function of the incident-reduced direction of the ray.