Slope sensitivities for optical surfaces

Setting a tolerance for the slope errors of an optical surface (e.g., surface form errors of the "mid-spatial-frequencies") requires some knowledge of how those surface errors affect the final image of the system. While excellent tools exist for simulating those effects on a surface-by-surface basis, considerable insight may be gained by examining, for each surface, a simple sensitivity parameter that relates the slope error on the surface to the ray displacement at the final image plane. Snell's law gives a relationship between the slope errors of a surface and the angular deviations of the rays emerging from the surface. For a singlet or thin doublet acting by itself, these angular deviations are related to ray deviations at the image plane by the focal length of the lens. However, for optical surfaces inside an optical system having a substantial axial extent, the focal length of the system is not the correct multiplier, as the sensitivity is influenced by the optical surfaces that follow. In this paper, a simple expression is derived that relates the slope errors at an arbitrary optical surface to the ray deviation at the image plane. This expression is experimentally verified by comparison to a real-ray perturbation analysis. The sensitivity parameter relates the RMS slope errors to the RMS spot radius, and also relates the peak slope error to the 100% spot radius, and may be used to create an RSS error budget for slope error. Application to various types of system are shown and discussed.