Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays

Snell's law describes the relationship between the incidence angle and reflection (or refraction) angle of a light ray impinging on the interface between two different isotropic media. In this paper, Snell's law is used to derive the unit normal vectors of an aspherical surface given a knowledge of the unit directional vectors of the incoming and outgoing rays. The proposed method has important applications in the design and fabrication of aspherical surfaces since the surface normal vectors determine not only the optical performance of the surface but also the cutting tool angles required to machine the surfaces.