Translations:Reflection and refraction – book/28/en

In the above derivations, the laws of reflection and refraction have been derived under the assumption that the incident wave is a plane wave and that the interface is a plane surface. However, the incident wave need not be a plane wave. It only needs to be a smooth curved surface - continuous and free of sharp bends or breaks. In such a case, we can concentrate our attention on an infinitesimal area of the wave surface, which can be considered plane. The two incident rays EA and FB then will be very close to each other and essentially will be parallel. Thus, the above derivations for finite wavefronts will hold also for infinitesimal wavefronts. The fact that these laws are stated in terms of geometric relationships between rays rather than between wavefronts removes the necessity for specifying that the wavefronts be either plane or infinitesimal in area. By similar reasoning, it can be argued that the reflecting or refracting surface need not necessarily be a plane surface but can be a smooth curved surface. A normal can be erected at any point on a smooth surface, so the various angles can be defined uniquely.