Refractive sphere and light point source: shortcut to the zero wavefronts.

A somewhat basic way to find an expression for the zero wavefront of a given illuminated refractive medium starts from a wavefront arbitrary point E, belonging to this medium, whose position analytical expression is already known. Then, one derives a new virtual wavefront-the zero wavefront-equivalent to the point source of light. The spatial path length of the resulting direct equivalent ray between E and the corresponding point E0, belonging to the zero wavefront, equals the optical path length of the more or less complicated succession of ray segments, caused by refraction and/or reflection, between E and the point source. Moreover, the ray direction of the equivalent direct ray, between E and E0, and that of the real ray at E must coincide. In the shortcut to the zero wavefront, one considers an arbitrary point E belonging rather to the entry interface of the optical medium and whose position analytical expression is already known. In the case of the refractive sphere illuminated by a point source, the internal progression of the ray implies, at each internal reflection point, two new media and two new zero wavefronts: one corresponding to the reflected fraction inside and the other corresponding to that refracted fraction outside. The analytical expression of the zero wavefront resulting from the shortcut, at least for the case of the refractive sphere, is not only much simpler, but as complete as the basic one. Indeed, the expression of any equivalent ray or wavefront can be obtained from the zero wavefront either through the basic way or through the shorter one.