Surface slope tolerances: the transition from geometric raytracing to scalar wave theory

In a previous presentation, the author presented a method of tolerancing mid-frequency surface ripple, based on a sensitivity parameter that related the slope error on the surface to the ray deviation at the image plane. The method established an upper allowable limit for the slope error on the surface. Because the analysis method was based on geometric raytracing, it included no consideration of scale. (Snell’s law is Snell’s law, regardless of the size of the feature.) Although the intention was to apply this design method only to macroscopic surface features, the notion of a slope limit without a scale limitation raises the question as to the region of validity, i.e., when do we need to take diffraction or scatter into account? In this presentation, we examine the transition between geometric raytracing and scalar diffraction theory. This is entirely analogous to the transition between the geometric model of a Ronchi test and that of a diffraction-based, lateral shear interferometer.