Supernumerary bows: caustics of a refractive sphere and analysis of the relative overall Gouy phase shift of supernumerary rays.

The modified Young's theory of interference related to supernumerary rainbows is based on a difference of 90° in the Gouy phase shifts for the parallel rays producing these bows. An observation screen placed at a given distance from a refractive sphere illuminated by a point source of light should also show supernumerary screen bows. An extensive description and analysis of the caustics involved are given. For any k order, k being the number of reflections inside the sphere, a procedure is established to determine the number of Gouy phase shifts encountered by any ray along its path from the source to the screen. Special consideration is given to the order k=0. For any k supernumerary bow, on any spherical screen whose center is that of the sphere, the difference in the Gouy phase shifts for the two rays producing a bow always amounts to 90°. An indirect proof of this characteristic is given. All considerations are made within the framework of geometrical optics being, on the one hand, the limit of the electromagnetic theory as the wavelength goes to 0, and being, on the other hand, complemented by the Gouy phase shift theory.